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•ISO 


HANDBOOK  OF  MATHEMATICS 
FOR  ENGINEERS 


%  Qraw~3/ill  Book  (n.  7ne 

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Handbook  of  Mathematics 
for  Engineers 


BY 


EDWARD  V.  HUNTINGTON,  PH.  D. 

ASSOCIATE    PROFESSOR    OF    MATHEMATICS,   HARVARD   UNIVERSITY 


WITH  TABLES  OF  WEIGHTS  AND  MEASURES  BY 


LOUIS  A.  FISCHER,  B 


CHIEF   OF   DIVISION   OF   WEIGHTS  AND 

U.   8.  BUREAU   OF  STANDARDS 


;.  S.  ^rrTT 

MEASURES,    4 


REPRINT  OF  SECTIONS  1  AND  2  OF  L.  S.  MARKS'S 
"MECHANICAL  ENGINEERS'  HANDBOOK" 


FIRST  EDITION 
SECOND  IMPRESSION 


McGRAW-HILL  BOOK  COMPANY,  INC, 

239  WEST  39TH  STREET.    NEW  YORK 


LONDON:  HILL  PUBLISHING  CO.,  LTD. 

6  &  8  BOUVERIE  ST.,  E.G. 
1918 


/ 

c 


COPYRIGHT,  1918,  BY  THE 
McGRAW  HILL  BOOK  COMPANY,  INC. 


COPYRIGHT,  1916,  BY 
EDWARD  V.  HUNTINGTON. 


TH»     M  A  F  L  1C     X>  R  IB  H  H     YOKJC     PA 


PREFACE 

This  Handbook  of  Mathematics  is  designed  to  contain,  in  compact  form, 
accurate  statements  of  those  facts  and  formulas  of  pure  mathematics  which 
are  most  likely  to  be  useful  to  the  worker  in  applied  mathematics. 

It  is  not  intended  to  take  the  place  of  the  larger  compendiums  of  pure 
mathematics  on  the  one  hand,  or  of  the  technical  handbooks  of  engineering 
on  the  other  hand;  but  in  its  own  field  it  is  thought  to  be  more  comprehensive 
than  any  other  similar  work  in  English. 

Many  topics  of  an  elementary  character  are  presented  in  a  form  which 
permits  of  immediate  utilization  even  by  readers  who  have  had  no  previous 
acquaintance  with  the  subject;  for  example,  the  practical  use  of  logarithms 
and  logarithmic  cross-section  paper,  and  the  elementary  parts  of  the  modern 
method  of  nomography  (alignment  charts),  can  be  learned  from  this  book 
without  the  necessity  of  consulting  separate  treatises. 

Other  sections  of  the  book  to  which  special  attention  may  be  called  are 
the  chapter  on  the  algebra  of  complex  (or  imaginary)  quantities,  the  treat- 
ment of  the  catenary  (with  special  tables),  and  the  brief  resume  of  the  theory 
of  vector  analysis. 

The  mathematical  tables  (including  several  which  are  not  ordinarily 
found)  are  carried  to  four  significant  figures  throughout,  and  no  pains  have 
been  spared  to  make  them  as  nearly  self-explanatory  as  possible,  even  to  the 
reader  who  makes  only  occasional  use  of  such  tables. 

For  the  Tables  of  Weights  and  Measures,  which  add  greatly  to  its  useful- 
ness, the  book  is  indebted  to  Mr.  Louis  A.  Fischer  of  the  U.  S.  Bureau  of 
Standards. 

All  the  matter  included  in  the  present  volume  was  originally  prepared  for 
the  Mechanical  Engineers'  Handbook  (Lionel  S.  Marks,  Editor-in-Chief), 
and  was  first  printed  in  1916,  as  Sections  1  and  2  of  that  Handbook.  The 
author  desires  to  express  his  indebtedness  to  Professor  Marks,  not  only  for 
indispensable  advice  as  to  the  choice  of  the  topics  which  would  be  most 
useful  to  engineers,  but  also  for  great  assistance  in  many  details  of  the 
presentation. 

All  the  misprints  that  have  been  detected  have  been  corrected  in  the  plates. 
Notification  in  regard  to  any  further  corrections,  and  any  suggestions  toward 
the  improvement  or  possible  enlargement  of  the  book,  will  be  cordially 
welcomed  by  the  author  or  the  publishers. 

E.  V.  H. 

CAMBRIDGE,  MASS. 
April  29,  1918. 


M171600 


CONTENTS 


Page 
PREFACE v 

SECTION  1.  Mathematical  Tables  and  Weights  and  Measures 1 

(For  detailed  Table  of  Contents,  see  page  1.) 

SECTION  2.  Mathematics: 

Arithmetic;  Geometry  and  Mensuration;  Algebra;  Trigonometry; 
Analytical  Geometry;  Differential  and  Integral  Calculus;  Graphical 
Representation  of  Functions;  Vector  Analysis 87 

(For  detailed  Table  of  Contents,  see  page  87.) 
INDEX  .  .    187 


SECTION  1 


MATHEMATICAL  TABLES 

AND 

WEIGHTS  AND  MEASURES 

BY 
EDWARD  V.  HUNTINGTON,  Ph.  D.,  Associate  Professor  of  Mathematics, 

Harvard  University,  Fellow  Am.  Acad.  Arts  and  Sciences. 
LOUISA.  FISCHER,  B.  S.,  Chief  of  Division  of  Weights  and  Measures, 

U.  S.  Bureau  of  Standards. 

CONTENTS 


MATHEMATICAL  TABLES 

BY  E.  V.  HUNTINGTON  PAGE 

Squares  of  Numbers 2 

Cubes  of  Numbers 8 

Square  Roots  of  Numbers 12 

Cube  Roots  of  Numbers 16 

Three-halves  Powers  of  Numbers. .  .  22 

Reciprocals  of  Numbers 24 

Circles     (Areas,     Segments,     etc.) .  28 

Spheres  (Volumes,  Segments,  etc.)..  36 

Regular  Polygons 39 

Binomial  Coefficients 39 

Common  Logarithms 40 

Degrees  and  Radians 44 

Trigonometric  Functions 46 

Exponentials 57 

Hyperbolic  (Napierian)  Logarithms.  58 

Hyperbolic  Functions 60 

Multiples  of  0.4343  and  2.3026 62 

Residuals  and  Probable  Errors 63 

Compound  Interest  and  Annuities.  64 

Decimal  Equivalents 69 


WEIGHTS  AND     MEASURES 

BY  LOUIS  A.  FISCHER        PAGE 
U.     S.     Customary     Weights     and 

Measures 70 

Metric  Weights  and  Measures 71 

Systems  of  Units 72 

Conversion  Tables: 

Lengths 74 

Areas 76 

Volumes  and  Capacities 76 

Velocities 78 

Masses  (Weights) 78 

Pressures 79 

Energy,  Work,  Heat 79 

Power 81 

Density 81 

Heat     Transmission     and      Con- 
duction   82 

Values  of  Foreign  Coins 82 

Time - 83 

Terrestrial  Gravity 84 

Specific  Gravity  and  Density 84 


MATHEMATICAL  TABLES 


SQUARES  OF 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

ll 

1.00 

1.000 

1.002 

1.004 

.006 

1.008 

1.010 

1.012 

1.014 

1.016 

1.018 

2 

1 

1.020 

1.022 

1.024 

.026 

1.028 

1.030 

1.032 

1.034 

1.036 

1.038 

2 

1.040 

1.042 

1.044 

.047 

1.049 

1.051 

1.053 

1.055 

1.057 

1.059 

3 

1.061 

1.063 

1.065 

.067 

1.069 

1.071 

1.073 

1.075 

1.077 

1.080 

4 

1.082 

1.084 

1.086 

.088 

1.090 

1.092 

1.094 

1.096 

1.098 

1.100 

1.05 

1.102 

1.105 

1.107 

.109 

1.111 

.113 

1.115 

1.117 

.119 

1.121 

6 

1.124 

1.126 

1.128 

.130 

1.132 

.134 

1.136 

1.138 

.141 

1.143 

7 

1.145 

1.147 

1.149 

.151 

1.153 

.156 

1.158 

1.160 

.162 

1.164 

8 

1.166 

1.169 

1.171 

.173 

1.175 

.177 

1.179 

1.182 

184 

1.186 

9 

1.188 

1.190 

1.192 

.195 

1.197 

.199 

1.201 

1.203 

.206 

1.208 

1.10 

1210 

1.212 

1.214 

.217 

1.219 

1.221 

1.223 

1.225 

.228 

1.230 

1 

1.232 

1.234 

1.237 

.239 

1.241 

1.243 

1.245 

1.248 

.250 

1.252 

2 

1.254 

1.257 

1.259 

.261 

1.263 

1.266 

1.268 

1.270 

.272 

1.275 

3 

1.277 

1.279 

1.281 

1.284 

1.286 

1.288 

1.209 

1.293 

.295 

1.297 

4 

1.300 

1.302 

1.304 

1.306 

1.309 

1.311 

1.313 

1.316 

.318 

1.320 

1.15 

1.322 

1.325 

1.327 

1.329 

1.332 

1.334 

1.336 

1.339 

.341 

1.343 

6 

1.346 

1.348 

1.350 

1353 

1.355 

1.357 

1.360 

1.362 

.364 

1.367 

7 

1.369 

1.371 

1.374 

1.376 

1.378 

1.381 

1.383 

1.385 

.388 

1.390 

8 

1.392 

1.395 

1.397 

1.399 

1.402 

1.404 

1.407 

1.409 

.411 

1.414 

9 

1.416 

1.418 

1.421 

1.423 

1.426 

1.428 

1.430 

1.433 

.435 

1.438 

1.20 

1.440 

1.442 

1.445 

1.447 

1.450 

1.452 

1.454 

1.457 

.459 

1.462 

1 

1.464 

1.467 

1.469 

1.471 

1.474 

1.476 

1.479 

1.481 

.484 

1.486 

2 

1.488 

1.491 

1.493 

1.496 

1.498 

1.501 

1.503 

1.506 

.508 

1.510 

3 

1.513 

1.515 

1.518 

1.520 

1.523 

1.525 

1.528 

1.530 

.533 

1.535 

4 

1.538 

1.540 

1.543 

1.545 

1.548 

1.550 

1.553 

1.555 

1.558 

1.560 

1.25 

1.562 

1.565 

1.568 

1.570 

1.573 

1.575 

1.578 

1.580 

1.583 

1585 

3 

6 

1.588 

1.590 

1.593 

1.595 

1.598 

1.600 

1.603 

1.605 

1.608 

1.610 

7 

1.613 

1.615 

1.618 

1.621 

1.623 

.626 

1.628 

1.631 

1.633 

1.636 

8 

1.638 

1.641 

1.644 

1.646 

1.649 

1.651 

1.654 

1.656 

1.659 

1.662 

9 

1.664 

1.667 

1.669 

1.672 

1.674 

1.677 

1.680 

1.682 

1.685 

1.687 

1.30 

1.690 

1.693 

1.695 

1.698 

1.700 

1.703 

1.706 

1.708 

1.711 

1.713 

1 

1.716 

1.719 

1.721 

1.724 

1.727 

1.729 

1.732 

1.734 

1.737 

1.740 

2 

1.742 

1.745 

1.748 

1.750 

1.753 

.756 

1.758 

1.761 

1.764 

1.766 

3 

1.769 

1.772 

1.774 

1.777 

1.780 

1.782 

1.785 

1.788 

1.790 

1.793 

4 

1.796 

1.798 

1.801 

1.804 

1.806 

1.809 

1.812 

1.814 

1.817 

1.820 

1.35 

1.822 

1.825 

1.828 

1.831 

1.833 

1.836 

1.839 

1.841 

1.844 

1.847 

6 

1.850 

1.852 

1.855 

1.858 

1.860 

1.863 

1.866 

1.869 

1.871 

1.874 

7 

1.877 

1.880 

1.882 

1.885 

1.888 

1.891 

1.893 

1.896 

1.899 

1.902 

8 

1.904 

1.907 

1.910 

1.913 

1.915 

1.918 

1.921 

1.924 

1.927 

1.929 

9 

1.932 

1.935 

1.938 

1.940 

1.943 

1.946 

1.949 

1.952 

1.954 

1.957 

1.40 

1.960 

1.963 

1.966 

1.968 

1.971 

1.974 

1.977 

1.980 

1.982 

1.985 

1 

1.988 

1.991 

1.994 

1.997 

1.999 

2.002 

2.005 

2.008 

2.011 

2.014 

2 

2.016 

2.019 

2.022 

2.025 

2.028 

2.031 

2.033 

2.036 

2.039 

2.042 

3 

2.045 

2.048 

2.051 

2.053 

2.056 

2.059 

2.062 

2.065 

2.068 

2.071 

4 

2.074 

2.076 

2.079 

2.082 

2.085 

2.088 

2.091 

2.094 

2.097 

2.100 

1.45 

2.102 

2.105 

2.108 

2.111 

2.114 

2.117 

2.120 

2.123 

2.126 

2.129 

6 

2.132 

2.135 

2.137 

2.140 

2.143 

2.146 

2.149 

2.152 

2.155 

2.158 

7 

2.161 

2.164 

2.167 

2.170 

2.173 

2.176 

2.179 

2.182 

2.184 

2.187 

8 

2.190 

2.193 

2.196 

2.199 

2.202 

2.205 

2.208 

2.211 

2.214 

2.217 

9 

2.220 

2.223 

2.226 

2.229 

2.232 

2.235 

2.238 

2.241 

2.244 

2.247 

Moving  the  decimal  point  ONE  place  in  N  requires  moving  it  TWO  places  in  body 
of  table  (see  p.  6). 


MATHEMATICAL  TABLES 


SQUARES  (continued) 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

ii 

1.50 

2.250 

2.253 

2.256 

2.259 

2.262 

2.265 

2.268 

2.271 

2.274 

2.277 

3 

I 

2.280 

2.283 

2.286 

2.289 

2.292 

2.295 

2.298 

2.301 

2.304 

2.307 

2 

2.310 

2.313 

2.316 

2.320 

2.323 

2.326 

2.329 

2.332 

2.335 

2.338 

3 

2.341 

2.344 

2.347 

2.350 

2.353 

2.356 

2.359 

2.362 

2.365 

2.369 

4 

2.372 

2.375 

2.378 

2.381 

2.384 

2.387 

2.390 

2.393 

2.396 

2.399 

1.55 

2.402 

2.406 

2.409 

2.412 

2.415 

2.418 

2.421 

2.424 

2.427 

2.430 

6 

2.434 

2.437 

2.440 

2.443 

2.446 

2.449 

2.452 

2.455 

2.459 

2.462 

7 

2.465 

2.468 

2.471 

2.474 

2.477 

2.481 

2.484 

2.487 

2.490 

2.493 

8 

2.496 

2.500 

2.503 

2.506 

2.509 

2.512 

2.515 

2.519 

2.522 

2.525 

9 

2.528 

2.531 

2.534 

2.538 

2.541 

2.544 

2.547 

2.550 

2.554 

2.557 

1.60 

2.560 

2.563 

2.566 

2.570 

2.573 

2.576 

2.579 

2.582 

2.586 

2.589 

1 

2.592 

2.595 

2.599 

2.602 

2.605 

2.608 

2.611 

2.615 

2.618 

2.621 

2 

2.624 

2.628 

2.631 

2.634 

2.637 

2.641 

2.644 

2.647 

2.650 

2.654 

3 

2.657 

2.660 

2.663 

2.667 

2.670 

2.673 

2.676 

2.680 

2.683 

2.686 

4 

2.690 

2.693 

2.696 

2.699 

2.703 

2.706 

2.709 

2.713 

2.716 

2.719 

1.65 

2.722 

2.726 

2.729 

2.732 

2.736 

2.739 

2.742 

2.746 

2.749 

2.752 

6 

2.756 

2.759 

2762 

2.766 

2.769 

2.772 

2.776 

2.779 

2.782 

2.786 

7 

2.789 

2.792 

2.796 

2.799 

2.802 

2.806 

2.809 

2.812 

2.816 

2.819 

8 

2.822 

2.826 

2.829 

2.832 

2.836 

2.839 

2.843 

2.846 

2849 

2.853 

9 

2.856 

2.859 

2.863 

2.866 

2.870 

2.873 

2.876 

2.880 

2.883 

2.887 

1.70 

2.890 

2.893 

2.897 

2.900 

2.904 

2.907 

2.910 

2.914 

2.917 

2.921 

1 

2.924 

2.928 

2.931 

2.934 

2.938 

2.941 

2.945 

2.948 

2.952 

2.955 

2 

2.958 

2.962 

2.965 

2.969 

2.972 

2.976 

2.979 

2.983 

2.986 

2.989 

3 

2.993 

2.996 

3.000 

3.003 

3.007 

3.010 

3.014 

3.017 

3.021 

3.024 

4 

3.028 

3.031 

3.035 

3.038 

3.042 

3.045 

3.049 

3.052 

3.056 

3.059 

1.75 

3.062 

3.066 

3.070 

3.073 

3.077 

3.080 

3.084 

3.087 

3.091 

3.094 

4 

6 

3.098 

3.101 

3.105 

3.108 

3.112 

3.115 

3.119 

3.122 

3.126 

3.129 

7 

3.133 

3.136 

3.140 

3.144 

3.147 

3.151 

3.154 

3.158 

3.161 

3.165 

.8 

3.168 

3.172 

3.176 

3.179 

3.183 

3.186 

3.190 

3.193 

3.197 

3-201 

9 

3.204 

3.208 

3.211 

3.215 

3.218 

3.222 

3.226 

3.229 

3.233 

3.236 

1.80 

3.240 

3.244 

3.247 

3.251 

3.254 

3.258 

3.262 

3.265 

3.269 

3.272 

1 

3.276 

3.280 

3.283 

3.287 

3.291 

3.294 

3.298 

3.301 

3.305 

3.309 

2 

3.312 

3.316 

3.320 

3.323 

3.327 

3.331 

3.334 

3.338 

3.342 

3.345 

3 

3.349 

3.353 

3.356 

3.360 

3.364 

3.367 

3.371 

3.375 

3.378 

3.382 

4 

3.386 

3.389 

3.393 

3.397 

3.400 

3.404 

3.408 

3.411 

3.415 

3.419 

1.85 

3.422 

3.426 

3.430 

3.434 

3.437 

3.441 

3.445 

3.448 

3.452 

3.456 

6 

3.460 

3.463 

3.467 

3.471 

3.474 

3.478 

3.482 

3.486 

3.489 

3.493 

7 

3.497 

3.501 

3.504 

3.508 

3.512 

3.516 

3.519 

3.523 

3.527 

3.531 

8 

3.534 

3.538 

3.542 

3.546 

3.549 

3.553 

3.557 

3.561 

3.565 

3.568 

9 

3.572 

3.576 

3.580 

3.583 

3.587 

3.591 

3.595 

3.599 

3.602 

3.606 

1.90 

3.610 

3.614 

3.618 

3.621 

3.625 

3.629 

3.633 

3.637 

3.640 

3.644 

1 

3.648 

3.652 

3.656 

3.660 

3.663 

3.667 

3.671 

3.675 

3.679 

3.683 

2 

3.686 

3.690 

3.694 

3.698 

3.702 

3.706 

3.709 

3.713 

3.717 

3.721 

3 

3.725 

3.729 

3.733 

3.736 

3.740 

3.744 

3.748 

3.752 

3.756 

3.760 

4 

3.764 

3.767 

3.771 

3.775 

3.779 

3.783 

3.787 

3.791 

3.795 

3.799 

1.95 

3.802 

3.806 

3.810 

3.814 

3.818 

3.822 

3.826 

3.830 

3.834 

3.838 

6 

3.842 

3.846 

3.849 

3.853 

3.857 

3.861 

3.865 

3.869 

3.873 

3.877 

7 

3.881 

3.885 

3.889 

3.893 

3.897 

3.901 

3.905 

3.909 

3.912 

3.916 

8 

3.920 

3.924 

3.928 

3.932 

3.936 

3.940 

3.944 

3.948 

3.952 

3.956 

9 

3.960 

3.964 

3.968 

3.972 

3.976 

3.980 

3.984 

3.988 

3.992 

3.996 

=  9.86960    !/*«  =  0.101321 


7.38906 


MATHEMATICAL  TABLES 


SQUARES  (continued) 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

$S 

2.00 

4.000 

4.004 

4.008 

4.012 

4.016 

4.020 

4.024 

4.028 

4.032 

4.036 

4 

1 

4.040 

4.044 

4.048 

4.052 

4.056 

4.060 

4.064 

4.068 

4.072 

4.076 

2 

4.080 

4.084 

4.088 

4.093 

4.097 

4.101 

4.105 

4.109 

4.113 

4.117 

3 

4.121 

4.125 

4.129 

4.133 

4.137 

4.141 

4.145 

4.149 

4.  153 

4.158 

4 

4.162 

4.166 

4.170 

4.174 

4.178 

4.182 

4.186 

4.190 

4.194 

4.198 

2.05 

4.202 

4.207 

4.211 

4.215 

4.219 

4.223 

4.227 

4.231 

4.235 

4.239 

6 

4.244 

4.248 

4.252 

4.256 

4.260 

4.264 

4.268 

4.272 

4.277 

4.281 

7 

4.285 

4.289 

4.293 

4.297 

4.301 

4.306 

4.310 

4.314 

4.318 

4.322 

8 

4.326 

4.331 

4.335 

4.339 

4.343 

4.347 

4.351 

4.356 

4.360 

4.364 

9 

4.368 

4.372 

4.376 

4.381 

4.385 

4.389 

4.393 

4.397 

4.402 

4.406 

2.10 

4.410 

4.414 

4.418 

4.423 

4.427 

4.431 

4.435 

4.439 

4.444 

4.448 

1 

4.452 

4.456 

4.461 

4.465 

4.469 

4.473 

4.477 

4.482 

4486 

4.490 

2 

4.494 

4.499 

4.503 

4.507 

4.511 

4.516 

4.520 

4.524 

4.528 

4533 

3 

4.537 

4.541 

4.545 

4.550 

4.554 

4.558 

4.562 

4.567 

4.571 

4.575 

4 

4.580 

4.584 

4.588 

4.592 

4.597 

4.601 

4.605 

4.610 

4.614 

4.618 

2.15 

4.622 

4.627 

4.631 

4.635 

4.640 

4.644 

4.648 

4.653 

4.657 

4.661 

6 

4.666 

4.670 

4.674 

4.679 

4.683 

4.687 

4.692 

4.696 

4.700 

4.705 

7 

4.709 

4.713 

4.718 

4.722 

4.726 

4.731 

4.735 

4.739 

4.744 

4.748 

8 

4.752 

4.757 

4.761 

4.765 

4.770 

4.774 

4.779 

4.783 

4.787 

4.792 

9 

4.796 

4.800 

4.805 

4.809 

4.814 

4.818 

4.822 

4.827 

4.831 

4.836 

2.20 

4.840 

4.844 

4.849 

4.853 

4.858 

4.862 

4.866 

4.871 

4.875 

4.880 

1 

4.884 

4.889 

4.893 

4.897 

4.902 

4.906 

4.911 

4.915 

4.920 

4.924 

2 

4.928 

4.933 

4.937 

4.942 

4.946 

4.951 

4.955 

4.960 

4.964 

4.968 

3 

4.973 

4.977 

4.982 

4.986 

4.991 

4.995 

5.000 

5.004 

5.009 

5.013 

4 

5.018 

5.022 

5.027 

5.031 

5.036 

5.040 

5.045 

5.049 

5.054 

5.058 

2.25 

5.062 

5.067 

5.072 

5.076 

5.081 

5.085 

5.090 

5.094 

5.099 

5.103 

5 

6 

5.108 

5.112 

5.117 

5.121 

5.126 

5.130 

5.135 

5.139 

5.144 

5148 

7 

5.153 

5.157 

5.162 

5.167 

5.171 

5.176 

5.180 

5.185 

5.189 

5.194 

8 

5.198 

5.203 

5.208 

5.212 

5.217 

5.221 

5.226 

5.230 

5.235 

5.240 

9 

5.244 

5.249 

5.253 

5.258 

5.262 

5.267 

5.272 

5.276 

5.281 

5.285 

2.30 

5.290 

5.295 

•5.299 

5.304 

5.308 

5.313 

5.318 

5.322 

5.327 

5.331 

1 

5.336 

5.341 

5.345 

5.350 

5.355 

5.359 

5.364 

5.368 

5.373 

5.378 

2 

5.382 

5.387 

5.392 

5.396 

5.401 

5.406 

5.410 

5.415 

5.420 

5.424 

3 

5.429 

5.434 

5.438 

5.443 

5.448 

5.452 

5.457 

5.462 

5.466 

5.471 

4 

5.476 

5.480 

5.485 

5.490 

5.494 

5.499 

5.504 

5.508 

5.513 

5.518 

2.35 

5.522 

5.527 

5.532 

5.537 

5.541 

5.546 

5.551 

5.555 

5.560 

5.565 

6 

5.570 

5.574 

5.579 

5.584 

5.588 

5.593 

5.598 

5.603 

5.607 

5.612 

7 

5.617 

5.622 

5.626 

5.631 

5.636 

5.641 

5.645 

5.650 

5.655 

5.660 

8 

5.664 

5.669 

5.674 

5.679 

5.683 

5.688 

5.693 

5.698 

5.703 

5.707 

9 

5.712 

5.717 

5.722 

5.726 

5.731 

5.736 

5.741 

5.746 

5.750 

5.755 

2.40 

5.760 

5.765 

5.770 

5.774 

5.779 

5.784 

5.789 

5.794 

5.798 

5.803 

1 

5.808 

5.813 

5.818 

5.823 

5.827 

5.832 

5.837 

5.842 

5.847 

5.852 

2 

5.856 

5.861 

5.866 

5.871 

5.876 

5.881 

5.885 

5.890 

5.895 

5.900 

3 

5.905 

5.910 

5.915 

5.919 

5.924 

5.929 

5.934 

5.939 

5.944 

5.949 

4 

5.954 

5.958 

5.963 

5.968 

5.973 

5.978 

5.983 

5.988 

5.993 

5.998 

2.45 

6.002 

6.007 

6.012 

6.017 

6.022 

6.027 

6.032 

6.037 

6.042 

6.047 

6 

6.052 

6.057 

6.061 

6.066 

6.071 

6.076 

6.081 

6.086 

6.091 

6.096 

7 

6.101 

6.106 

6.111 

6.116 

6.121 

6.126 

6.131 

6.136 

6.140 

6.145 

8 

6.150 

6.155 

6.160 

6.165 

6.170 

6.175 

6.180 

6.185 

6.190 

6.195 

9 

6.200 

6.205 

6.210  , 

6.215 

6.220 

6.225 

6.230 

6.235 

6.240 

6.245 

Moving  the  decimal  point  ONE  place  in  N  requires  moving  it  TWO  places  in  body 
of  table  (see  p.  G). 


MATHEMATICAL  TABLES 


SQUARES   (continued) 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

£ 

2.50 

6.250 

6.255 

6.260 

6.265 

6.270 

6.275 

6.280 

6.285 

6.290 

6.295 

5 

1 

6.300 

6.305 

6.310 

6.315 

6.320 

6.325 

6.330 

6.335 

6.340 

6.345 

2 

6.350 

6.355 

6.360 

6.366 

6.371 

6.376 

6.381 

6.386 

6.391 

6.396 

3 

6.401 

6.406 

6.411 

6.416 

6.421 

6.426 

6.431 

6.436 

6.441 

6.447 

4 

6.452 

6.457 

6.462 

6.467 

6.472 

6.477 

6.482 

6.487 

6.492 

6.497 

2.55 

6.502 

6.508 

6.513 

6.518 

6.523 

6.528 

6.533 

6.538 

6.543 

6.548 

6 

6.554 

6.559 

6.564 

6.569 

6.574 

6.579 

6.584 

6.589 

6.595 

6.600 

7 

6.605 

6.610 

6.615 

6.620 

6.625 

6.631 

6.636 

6.641 

6.646 

6.651 

8 

6.656 

6.662 

6.667 

6.672 

6.677 

6.682 

6.687 

6.693 

6.698 

6.703 

9 

6.708 

6.713 

6.718 

6.724 

6.729 

6.734 

6.739 

6.744 

6.750 

6.755 

2.60 

6.760 

6.765 

6.770 

6.776 

6.781 

6.786 

6.791 

6.7% 

6.802 

6.807 

1 

6.812 

6.817 

6.823 

6.828 

6.833 

6.838 

6.843 

6.849 

6.854 

6.859 

2 

6.864 

6.870 

6.875 

6.880 

6.885 

6.891 

6.896 

6.901 

6906 

6.912 

3 

6.917 

6.922 

6.927 

6.933 

6.938 

6.943 

6.948 

6.954 

6.959 

6.964 

4 

6.970 

6.975 

6.980 

6.985 

6.991 

6.996 

7.001 

7.007 

7.012 

7.017 

2.65 

7.022 

7.028 

7.033 

7.038 

7.044 

7.049 

7.054 

7.060 

7.065 

7.070 

6 

7.076 

7.081 

7.086 

7.092 

7.097 

7.102 

7.108 

7.113 

7.118 

7.124 

7 

7.129 

7.134 

7.140 

7.145 

7.150 

7.156 

7.161 

7.166 

7.172 

7.177 

8 

7.182 

7.188 

7.193 

7.198 

7.204 

7.209 

7.215 

7.220 

7.225 

7.231 

9 

7.236 

7.241 

7.247 

7.252 

7.258 

7.263 

7.268 

7.274 

7.279 

7.285 

2.70 

7.290 

7.295 

7.301 

7.306 

7.312 

7.317 

7.322 

7.328 

7.333 

7339 

1 

7.344 

7.350 

7.355 

7.360 

7.366 

7.371 

7.377 

7.382 

7.388 

7.393 

2 

7.398 

7.404 

7.409 

7.415 

7.420 

7.426 

7.431 

7.437 

7.442 

7.447 

3 

7.453 

7.458 

7.464 

7.469 

7.475 

7.480 

7.486 

7.491 

7.497 

7.502 

4 

7.508 

7.513 

7.519 

7.524 

7.530 

7.535 

7.541 

7.546 

7.552 

7.557 

2.75 

7.562 

7.568 

7.574 

7.579 

7.585 

7.590 

7.596 

7.601 

7.607 

7.612 

6 

6 

7.618 

7.623 

7.629 

7.634 

7.640 

7.645 

7.651 

7.656 

7.662 

7.667 

7 

7.673 

7.678 

7.684 

7.690 

7.695 

7.701 

7.706 

7.712 

7.717 

7.723 

8 

7.728 

7.734 

7.740 

7.745 

7.751 

7.756 

7.762 

7.767 

7.773 

7.779 

9 

7.784 

7.790 

7.795 

7.801 

7.806 

7.812 

7.818 

7.823 

7.829 

7.834 

2.80 

7.840 

7.846 

7.851 

7.857 

7.862 

7.868 

7.874 

7.879 

7.885 

7.890 

1 

7.896 

7.902 

7.907 

7.913 

7.919 

7.924 

7.930 

7.935 

7.941 

7.947 

2 

7.952 

7.958 

7.964 

7.969 

7.975 

7.981 

7.986 

7.992 

7.998 

8003 

3 

8.009 

8.015 

8.020 

8.026 

8.032 

8.037 

8.043 

8.049 

8.054 

8.060 

4 

8.066 

8.071 

8.077 

8.083 

8.088 

8.094 

8.100 

8.105 

8.111 

8.117 

2.85 

8.122 

8.128 

8.134 

8.140 

8.145 

8.151 

8.157 

8.162 

8.168 

8.174 

6 

8.180 

8.185 

8.191 

8.197 

8.202 

8.208 

8.214 

8.220 

8.225 

8.231 

7 

8.237 

8.243 

8.248 

8.254 

8.260 

8.266 

8.271 

8.277 

8.283 

8.289 

8 

8.294 

8.300 

8.306 

8.312 

8.317 

8.323 

8.329 

8.335 

8.341 

8.346 

9 

8.352 

8.358 

8.364 

8.369 

8.375 

8.381 

8.387 

8.393 

8.398 

8.404 

2.90 

8.410 

8.416 

8.422 

8.427 

8.433 

8.439 

8.445 

8.451 

8.456 

8.462 

1 

8.468 

8.474 

8.480 

8.486 

8.491 

8.497 

8.503 

8.509 

8.515 

8.521 

2 

8526 

8.532 

8.538 

8.544 

8.550 

8.556 

8.561 

8.567 

8.573 

8.579 

3 

8.585 

8.591 

8.597 

8.602 

8.608 

8.614 

8.620 

8.626 

8.632 

8.638 

4 

8.644 

8.649 

8.655 

8.661 

8.667 

8.673 

8.679 

8.685 

8.691 

8.697 

2.95 

8.702 

8.708 

8.714 

8.720 

8.726 

8.732 

8.738 

8.744 

8.750 

8.756 

6 

8.762 

8.768 

8.773 

8.779 

8.785 

8.791 

8.797 

8.803 

8.809 

8.815 

7 

8.821 

8.827 

8.833 

8.839 

8.845 

8.851 

8.857 

8.863 

8.868 

8874 

8 

8.880 

8.886 

8.892 

8.898 

8.904 

8.910 

8.916 

8.922 

8.928 

8.934 

9 

8.940 

8.946 

8.952 

8.958 

8.964 

8.970 

8.976 

8.982 

8.988 

8.994 

I/T»  =  0.101321 


7.38906 


c 


MATHEMATICAL  TABLES 


SQUARES  (continued} 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

S?«a 

•5* 

3.00 

9.000 

9.006 

9.012 

9.018 

9.024 

9.030 

9.036 

9.042 

9.048 

9.054 

6 

1 

9.060 

9.066 

9.072 

9.078 

9.084 

9.090 

9.096 

9.102 

9.108 

9.114 

2 

9.120 

9.126 

9.132 

9.139 

9.145 

9.151 

9.157 

9.163 

9.169 

9.175 

3 

9.181 

9.187 

9.193 

9.199 

9.205 

9.211 

9.217 

9.223 

9.229 

9.236 

4 

9.242 

9.248 

9.254 

9.260 

9.266 

9.272 

9.278 

9.284 

9.290 

9.296 

3.05 

9.302 

9.309 

9.315 

9.321 

9.327 

9.333 

9.339 

9.345 

9.351 

9.357 

6 

9.364 

9.370 

9.376 

9.382 

9.388 

9.394 

9.400 

9.406 

9.413 

9.419 

7 

9.425 

9.431 

9.437 

9.443 

9.449 

9.456 

9.462 

9.468 

9.474 

9.480 

8 

9.486 

9.493 

9.499 

9.505 

9.511 

9.517 

9.523 

9.530 

9.536 

9.542 

9 

9.548 

9.554 

9.560 

9.567 

9.573 

9.579 

9.585 

9.591 

9.598 

9.604 

3.10 

9.610 

9.616 

9.622 

9.629 

9.635 

9.641 

9.647 

9.653 

9.660 

9.666 

1 

9.672 

9.678 

9.685 

9.691 

9.697 

9.703 

9.709 

9.716 

9.722 

9.728 

2 

9.734 

9.741 

9.747 

9.753 

9.759 

9.766 

9.772 

9.778 

9.784 

9.791 

3 

9.797 

9.803 

9.809 

9.816 

9.822 

9.828 

9.834 

9.841 

9.847 

9.853 

4 

9.860 

9.866 

9.872 

9.878 

9.885 

9.891 

9.897 

9.904 

9.910 

9.916 

3.15 

9.922 

9.929 

9.935 

9.941 

9.948 

9.954 

9.960 

9.967 

9.973 

9.979 

6 

9.986 

9.992 

9.998 

10.005 

6 

3.1 

9.99 

10.05 

10.11 

10.18 

6 

2 

10.24 

10.30 

10.37 

10.43 

10.50 

10.56 

10.63 

10.69 

10.76 

10.82 

3 

10.89 

10.96 

11.02 

11.09 

11.16 

11.22 

11.29 

11.36 

11.42 

11.49 

7 

4 

11.56 

11.63 

11.70 

11.76 

11.83 

11.90 

11.97 

12.04 

12.11 

12.18 

3.5 

12.25 

12.32 

12.39 

12.46 

12.53 

12.60 

12.67 

12.74 

12.82 

12.89 

6 

12.96 

13.03 

13.10 

13.18 

13.25 

13.32 

13.40 

13.47 

13.54 

13.62 

7 

13.69 

13.76 

13.84 

13.91 

13.99 

14.06 

14.14 

14.21 

14.29 

14.36 

8 

8 

14.44 

14.52 

14.59 

14.67 

14.75 

14.82 

14.90 

14.98 

15.05 

15.13 

9 

15.21 

15.29 

15.37 

15.44 

15.52 

15.60 

15.68 

15.76 

15.84 

15.92 

4.0 

16.00 

16.08 

16.16 

16.24 

16.32 

16.40 

16.48 

16.56 

16.65 

16.73 

1 

16.81 

16.89 

16.97 

17.06 

17.14 

17.22 

17.31 

17.39 

17.47 

17.56 

2 

17.64 

17.72 

17.81 

17.89 

17.98 

18.06 

18.15 

18.23 

18.32 

18.40 

3 

18.49 

18.58 

18.66 

18.75 

18.84 

18.92 

19.01 

19.10 

19.18 

19.27 

9 

4 

19.36 

19.45 

19.54 

19.62 

19.71 

19.80 

19.89 

19.98 

20.07 

20.16 

4.5 

20.25 

20.34 

20.43 

20.52 

20.61 

20.70 

20.79 

20.88 

20.98 

21.07 

6 

21.16 

21.25 

21.34 

21.44 

21.53 

21.62 

21.72 

21.81 

21.90 

22.00 

7 

22.09 

22.18 

22.28 

22.37 

22.47 

22.56 

22.66 

22.75 

22.85 

22.94 

10 

8 

23.04 

23.14 

23.23 

23.33 

23.43 

23.52 

23.62 

23.72 

23.81 

23.91 

9 

24.01 

24.11 

24.21 

24.30 

24.40 

24.50 

24.60 

24.70 

24.80 

24.90 

9.86960   (x/2)2  =  2.46740   !/«•»  =  0.101321 


Explanation  of  Table  of  Squares  (pp.  2-7). 

This  table  gives  the  value  of  Nz  for  values  of  N  from  1  to  10,  correct  to  four  figures. 
(Interpolated  values  may  be  in  error  by  1  in  the  fourth  figure). 

To  find  the  square  of  a  number  N  outside  the  range  from  1  to  10,  note  that 
moving  the  decimal  point  one  place  in  column  N  is  equivalent  to  moving  it  two  places 
in  the  body  of  the  table.  For  example: 

(3.217)2  -  10.35;        (0.03217)2  =  0.001035;        (3217)*  =  10350000 

This  table  can  also  be  used  inversely,  to  give  square  roots. 


MATHEMATICAL  TABLES 


SQUARES  (continued) 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

& 

5.0 

25.00 

25.10 

25.20 

25.30 

25.40 

25.50 

25.60 

25.70 

25.81 

25.91 

10 

1 

26.01 

26.11 

26.21 

26.32 

26.42 

26.52 

26.63 

26.73 

26.83 

26.94 

2 

27.04 

27.14 

27.25 

27.35 

27.46 

27.56 

27.67 

27.77 

27.88 

27.98 

3 

28.09 

28.20 

28.30 

28.41 

28.52 

28.62 

28.73 

28.84 

28.94 

29.05 

11 

4 

29.16 

29.27 

29.38 

29.48 

29.59 

29.70 

29.81 

29.92 

30.03 

30.14 

5.5 

30.25 

30.36 

30.47 

30.58 

30.69 

30.80 

30.91 

31.02 

31.14 

31.25 

6 

31.36 

31.47 

31.58 

31.70 

31.81 

31.92 

32.04 

32.15 

32.26 

32.38 

7 

32.49 

32.60 

32.72 

32.83 

32.95 

33.06 

33.18 

33.29 

33.41 

33.52 

8 

33.64 

33.76 

33.87 

33.99 

34.11 

34.22 

34.34 

34.46 

34.57 

34.69 

12 

9 

34.81 

34.93 

35.05 

35.16 

35.28 

35.40 

35.52 

35.64 

35.76 

35.88 

6.0 

36.00 

36.12 

36.24 

36.36 

36.48 

36.60 

36.72 

36.84 

36.97 

37.09 

37.21 

37.33 

37.45 

37.58 

37.70 

37.82 

37.95 

38.07 

38.19 

38.32 

2 

38.44 

38.56 

38.69 

38.81 

38.94 

39.06 

39.19 

39.31 

39.44 

39.56 

3 

39.69 

39.82 

39.94 

40.07 

40.20 

40.32 

40.45 

40.58 

40.70 

40.83 

13 

4 

40.96 

41.09 

41.22 

41.34 

41.47 

41.60 

41.73 

41.86 

41.99 

42.12 

6.5 

42.25 

42.38 

42.51 

42.64 

42.77 

42.90 

43.03 

43.16 

43.30 

43.43 

6 

43.56 

43.69 

43.82 

43.96 

44.09 

44.22 

44.36 

44.49 

44.62 

44.76 

7 

44.89 

45.02 

45.16 

45.29 

45.43 

45.56 

45.70 

45.83 

45.97 

46.10 

8 

46.24 

46.38 

46.51 

46.65 

46.79 

46.92 

47.06 

47.20 

47.33 

47.47 

14 

9 

47.61 

47.75 

47.89 

48.02 

48.16 

48.30 

48.44 

48.58 

48.72 

48.86 

7.0 

49.00 

49.14 

49.28 

49.42 

49.56    * 

49.70 

49.84 

49.98 

50.13 

50.27 

1 

50.41 

50.55 

50.69 

50.84 

50.98 

51.12 

51.27 

51.41 

51.55 

51.70 

2 

51.84 

51.98 

52.13 

52.27 

52.42 

52.56 

52.71 

52.85 

53.00 

53.14 

3 

53.29 

53.44 

53.58 

53.73 

53.88 

54.02 

54.17 

54.32 

54.46 

54.61 

15 

4 

54.76 

54.91 

55.06 

55.20 

55.35 

55.50 

55.65 

55.80 

55.95 

56.10 

7.5 

56.25 

56.40 

56.55 

56.70 

56.85 

57.00 

57.15 

57.30 

57.46 

57.61 

6 

57.76 

57.91 

58.06 

58.22 

58.37 

58.52 

58.68 

58.83 

58.98 

59.14 

7 

59.29 

59.44 

59.60 

59.75 

59.91 

60.06 

60.22 

60.37 

60.53 

60.68 

8 

60.84 

61.00 

61.15 

61.31 

61.47 

61.62 

61.78 

61.94 

62.09 

62.25 

16 

9 

62.41 

62.57 

62.73 

62.88 

63.04 

63.20 

63.36 

63.52 

63.68 

63.84 

8.0 

64.00 

64.16 

64.32 

64.48 

64.64 

64.80 

64.96 

65.12 

65.29 

65.45 

] 

65.61 

65.77 

65.93 

66.10 

66.26 

66.42 

66.59 

66.75 

66.91 

67.08 

2 

67.24 

67.40 

67.57 

67.73 

67.90 

68.06 

68.23 

68.39 

68.56 

68.72 

3 

68.89 

69.06 

69.22 

69.39 

69.56 

69.72 

69.89 

70.06 

70.22 

70.39 

17 

4 

70.56 

70.73 

70.90 

71.06 

71.23 

71.40 

71.57 

71.74 

71.91 

72.08 

8.5 

72.25 

72.42 

72.59 

72.76 

72.93 

73.10 

73.27 

73.44 

73.62 

73.79 

6 

73.96 

74.13 

74.30 

74.48 

74.65 

74.82 

75.00 

75.17 

75.34 

75.52 

7 

75.69 

75.86 

76.04 

76.21 

76.39 

76.56 

76.74 

76.91 

77.09 

77.26 

8 

77.44 

77.62 

77.79 

77.97 

78.15 

78.32 

78.50 

78.68 

78.85 

79.03 

18 

9 

79.21 

79.39 

79.57 

79.74 

79.92 

80.10 

80.28 

80.46 

80.64 

80.82 

9.0 

81.00 

81.18 

81.36 

81.54 

81.72 

81.90 

82.08 

82.26 

82.45 

82.63 

1 

82.81 

82.99 

83.17 

83.36 

83.54 

83.72 

83.91 

84.09 

84.27 

84.46 

2 

84.64 

84.82 

85.01 

85.19 

85.38 

85.56 

85.75 

85.93 

86.12 

86.30 

3 

86.49 

86.68 

86.86 

87.05 

87.24 

87.42 

87.61 

87.80 

87.98 

88.17 

19 

4 

88.36 

88.55 

88.74 

88.92 

89.11 

89.30 

89.49 

89.68 

89.87 

90.06 

9.5 

90.25 

90.44 

90.63 

90.82 

91.01 

91.20 

91.39 

91.58 

91.78 

91.97 

6 

92.16 

92.35 

92.54 

92.74 

92.93 

93.12 

93.32 

93.51 

93.70 

93.90 

7 

94.09 

94.28 

94.48 

94.67 

94.87 

95.06 

95.26 

95.45 

95.65 

95.84 

8 

96.04 

96.24 

96.43 

96.63 

96.83 

97.02 

97.22 

97.42 

97.61 

97.81 

20 

9 

9801 

98.21 

98.41 

98.60 

98.80 

99.00 

99.20 

99.40 

99.60 

99.80 

10.0 

100.0 

Moving  the  decimal  point  ONE  place  in  N  requires  moving  it  TWO  places  in  body 
of  table  (see  p.  6). 


MATHEMATICAL  TABLES 


CUBES  OP  NUMBERS 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

II 

1.00 

1.000 

1.003 

1.006 

1.009 

1.012 

1.015 

1.018 

1/»21 

1.024 

1.027 

3 

1 

1.030 

1.033 

1.036 

1.040 

1.043 

1.046 

1.049 

1.052 

1.055 

1.058 

2 

1.061 

1.064 

1.067 

1.071 

1.074 

1.077 

1.080 

1.083 

1.086 

1.090 

3 

1.093 

1.096 

1.099 

1.102 

1.106 

1.109 

1.112 

1.115 

1.118 

1.122 

4 

1.125 

1.128 

1.131 

1.135 

1.138 

1.141 

1.144 

1.148 

1.151 

1.154 

1.05 

1.158 

.161 

1.164 

1.168 

1.171 

1.174 

1.178 

1.181 

1.184 

1.188 

6 

1.191 

.194 

1.198 

1.201 

1.205 

1.208 

1.211 

1.215 

1.218 

1.222 

7 

1.225 

.228 

1.232 

1.235 

1.239 

1.242 

1.246 

1.249 

1.253 

1.256 

8 

1.260 

.263 

1.267 

1.270 

1.274 

1.277 

1.281 

1.284 

1.288 

1.291 

4 

9 

1.295 

.299 

1.302 

1.306 

1.309 

1.313 

1.317 

1.320 

1.324 

1.327 

1.10 

1.331 

.335 

1.338 

1.342 

1.346 

1.349 

1.353 

1.357 

1.360 

1.364 

1 

1368 

.371 

1.375 

1.379 

1.382 

1.386 

1.390 

1.394 

1.397 

1.401 

2 

1.405 

1.409 

1.412 

1.416 

1.420 

1.424 

1.428 

1.431 

1.435 

1.439 

3 

1.443 

1.447 

1.451 

1.454 

1.458 

1.462 

1.466 

1.470 

1.474 

1.478 

4 

1.482 

1.485 

1.489 

1.493 

1.497 

1.501 

1.505 

1.509 

1.513 

1.517 

1.15 

1.521 

1.525 

1.529 

1.533 

1.537 

1.541 

1.545 

1.549 

1.553 

1.557 

6 

1.561 

1.565 

1.569 

1.573 

1.577 

1.581 

1.585 

1.589 

1.593 

1.598 

7 

1.602 

1.606 

1.610 

1.614 

1.618 

1.622 

1.626 

1.631 

1.635 

1.639 

8 

1.643 

1.647 

1.651 

1.656 

1.660 

1.664 

1.668 

1.672 

1.677 

1.681 

9 

1.685 

1.689 

1.694 

1.698 

1.702 

1.706 

1.711 

1.715 

1.719 

1.724 

1.20 

1.728 

1.732 

1.737 

1.741 

1.745 

1.750 

1.754 

1.758 

1.763 

1.767 

1 

1.772 

1.776 

1.780 

1.785 

1.789 

1.794 

1.798 

1.802 

1.807 

1.811 

2 

1.816 

1.820 

1.825 

1.829 

1.834 

1.838 

1.843 

1.847 

1.852 

1.856 

3 

1.861 

1.865 

1.870 

1.875 

1.879 

1.884 

1.888 

1.893 

1.897 

1.902 

4 

1.907 

1.911 

1.916 

1.920 

1.925 

1.930 

1.934 

1.939 

1.944 

1.948 

5 

1.25 

1.953 

1.958 

1.963 

1.967 

1.972 

1.977 

1.981 

1.986 

1.991 

1.996 

6 

2.000 

2.005 

2.010 

2.015 

2.019 

2.024 

2.029 

2.034 

2.039 

2.044 

7 

2.048 

2.053 

2.058 

2.063 

2.068 

2.073 

2.078 

2.082 

2.087 

2.092 

8 

2.097 

2.102 

2.107 

2.112 

2.117 

2.122 

2.127 

2.132 

2.137 

2.142 

9 

2.147 

2.152 

2.157 

2.162 

2.167 

2.172 

2.177 

2.182 

2.187 

2.192 

1.30 

2.197 

2.202 

Z.207 

/.212 

2.217 

2.222 

2.228 

2.233 

2.238 

2.243 

1 

2.248 

2.253 

2.258 

2.264 

2.269 

2.274 

2.279 

2.284 

2.290 

2.295 

2 

2.300 

2.305 

2.310 

2.316 

2.321 

2.326 

2.331 

2.337 

2.342 

2.347 

3 

2.353 

2.358 

2.363 

2.369 

2.374 

2.379 

2.385 

2.390 

2.395 

2.401 

4 

2.406 

2.411 

2.417 

2.422 

2.428 

2.433 

2.439 

2.444 

2.449 

2.455 

1.35 

2.460 

2.466 

2.471 

2.477 

2.482 

2.488 

2.493 

2.499 

2.504 

2.510 

6 

6 

2.515 

2.521 

2.527 

2.532 

2.538 

2.543 

2.549 

2.554 

2.560 

2.566 

7 

2,571 

2.577 

2.583 

2.588 

2.594 

2.600 

2.605 

2.611 

2.617 

2.622 

8 

2.628 

2.634 

2.640 

2.645 

2.651 

2.657 

2.663 

2.668 

2.674 

2.680 

9 

2.686 

2.691 

2.697 

2.703 

2.709 

2.715 

2.721 

2.726 

2.732 

2.738 

1.40 

2.744 

2.750 

2.756 

2.762 

2.768 

2.774 

2.779 

2.785 

2.791 

2.797 

] 

2.803 

2.809 

2.815 

2.821 

2.827 

2.833 

2.839 

2.845 

2.851 

2.857 

2 

2.863 

2.869 

2.875 

2.881 

2.888 

2.894 

2.900 

2.906 

2.912 

2.918 

3 

2.924 

2.930 

2.936 

2.943 

2.949 

2.955 

2.961 

2.967 

2.974 

2.980 

4 

2.986 

2.992 

2.998 

3.005 

3.011 

3.017 

3.023 

3.030 

3.036 

3.042 

1.45 

3.049 

3.055 

3.061 

3.068 

3.074 

3.080 

3.087 

3.093 

3.099 

3.106 

6 

3.112 

3.119 

3.125 

3.131 

3.138 

3.144 

3.151 

3.157 

3.164 

3.170 

7 

3.177 

3.183 

3.190 

3.196 

3.203 

3.209 

3.216 

3.222 

3.229 

3.235 

8 

3.242 

3.248 

3.255 

3.262 

3.268 

3.275 

3.281 

3.288 

3.295 

3.301 

7 

9 

3.308 

3.315 

3.321 

3.328 

3.335 

3.341 

3.348 

3.355 

3.362 

3.368 

Moving  the  decimal  point  ONE  place  in  N  requires   moving  it   THREE   places  in 
body  of  table  (see  p.  10). 


MATHEMATICAL   TABLES 


CUBES   (continued) 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

& 

1.50 

3.375 

3.382 

3.389 

3.395 

3.402 

3.409 

3.416 

3.422 

3.429 

3.436 

7 

1 

3.443 

3.450 

3.457 

3.464 

3.470 

3.477 

3.484 

3.491 

3.498 

3.505 

2 

3.512 

3.519 

3.526 

3.533 

3.540 

3.547 

3.554 

3.561 

3.568 

3.575 

3 

3.582 

3.589 

3.596 

3.603 

3.610 

3.617 

3.624 

3.631 

3.638 

3.645 

4 

3.652 

3.659 

3.667 

3.674 

3.681 

3.688 

3.695 

3.702 

3.709 

3.717 

1.55 

3.724 

3.731 

3.738 

3.746 

3.753 

3.760 

3.767 

3.775 

3.782 

3.789 

6 

3.796 

3.804 

3.811 

3.818 

3.826 

3.833 

3.840 

3.848 

3.855 

3.863 

7 

3.870 

3.877 

3.885 

3.892 

3.900 

3.907 

3.914 

3.922 

3.929 

3.937 

8 

3.944 

3.952 

3.959 

3.967 

3.974 

3.982 

3.989 

3.997 

4.005 

4.012 

8 

9 

4.020 

4.027 

4.035 

4.042 

4.050 

4.058 

4.065 

4.073 

4.081 

4.088 

1.60 

4.096 

4.104 

4.111 

4.119 

4.127 

4.135 

4.142 

4.150 

4.158 

4.166 

4.173 

4.181 

4.189 

4.197 

4.204 

4.212 

4.220 

4.228 

4.236 

4.244 

2 

4.252 

4.259 

4.267 

4.275 

4.283 

4.291 

4.299 

4.307 

4.315 

4.323 

3 

4.331 

4.339 

4.347 

4355 

4.363 

4.371 

4.379 

4.387 

4.395 

4.403 

4 

4.411 

4.419 

4.427 

4.435 

4.443 

4.451 

4.460 

4.468 

4.476 

4.484 

1.65 

4.492 

4.500 

4.508 

4.517 

4.525 

4.533 

4.541 

4.550 

4.558 

4.566 

6 

4.574 

4.583 

4.591 

4.599 

4.607 

4.616 

4.624 

4.632 

4.641 

4.649 

7 

4.657 

4.666 

4.674 

4.683 

4.691 

4.699 

4.708 

4.716 

4.725 

4.733 

8 

4.742 

4.750 

4.759 

4.767 

4.776 

4.784 

4.793 

4.801 

4.810 

4.818 

9 

4.827 

4.835 

4.844 

4.853 

4.861 

4.870 

4.878 

4.887 

4.896 

4.904 

9 

1.70 

4.913 

4.922 

4.930 

4.939 

4.948 

4.956 

4.965 

4.974 

4.983 

4.991 

1 

5.000 

5.009 

5.018 

5.027 

5.035 

5.044 

5.053 

5.062 

5.071 

5.080 

2 

5.088 

5.097 

5.106 

5.115 

5.124 

5.133 

5.142 

5.151 

5.160 

5.169 

3 

5.178 

5.187 

5.196 

5.205 

5.214 

5.223 

5.232 

5.241 

5.250 

5.259 

4 

5.268 

5.277 

5.286 

5.295 

5.304 

5.314 

5.323 

5.332 

5.341 

5.350 

1.75 

5.359 

5.369 

5.378 

5.387 

5.396 

5.405 

5.415 

5.424 

5.433 

5.442 

6 

5.452 

5.461 

5.470 

5.480 

5.489 

5.498 

5.508 

5.517 

5.526 

5.536 

7 

5.545 

5.555 

5.564 

5.573 

5.583 

5.592 

5.602 

5.611 

5.621 

5.630 

10 

8 

5.640 

5.649 

5.659 

5.668 

5.678 

5.687 

5.697 

5.707 

5,716 

5.726 

9 

5.735 

5.745 

5.755 

5.764 

5.774 

5.784 

5.793 

5.803 

5.813 

5.822 

1.80 

5.832 

5.842 

5.851 

5.861 

5.871 

.5.881 

5.891 

5.900 

5.910 

5.920 

1 

5.930 

5.940 

5.949 

5.959 

5.969 

5.979 

5.989 

5.999 

6.009 

6.019 

2 

6.029 

6.039 

6.048 

6.058 

6.068 

6.078 

6.088 

6.098 

6.108 

6.118 

3 

6.128 

6.139 

6.149 

6.159 

6.  169 

6.179 

6.189 

6.199 

6.209 

6.219 

4 

6.230 

6.240 

6.250 

6.260 

6.270 

6.280 

6.291 

6.301 

6.311 

6.321 

1.85 

6.332 

6.342 

6.352 

6.362 

6.373 

6.383 

6.393 

6.404 

6.414 

6.424 

6 

6.435 

6.445 

6.456 

6.466 

6.476 

6.487 

6.497 

6.508 

6.518 

6.529 

7 

6.539 

6.550 

6.560 

6.571 

6.581 

6.592 

6.602 

6.613 

6.623 

6.634 

11 

8 

6.645 

6.655 

6.666 

6.677 

6.687 

6.698 

6.708 

6.719 

6.730 

6.741 

9 

6.751 

6.762 

6.773 

6.783 

6.794 

6.805 

6.816 

6.827 

6.837 

6.848 

1.90 

6.859 

6.870 

6.881 

6.892 

6.902 

6.913 

6.924 

6.935 

6.946 

6.957 

1 

6.968 

6.979 

6.990 

7.001 

7.012 

7.023 

7.034 

7.045 

7.056 

7.067 

2 

7.078 

7.089 

7.100 

7.1  11 

7.122 

7.133 

7.144 

7.156 

7.167 

7.178 

3 

7.189 

7.200 

7.211 

7.223 

7.234 

7.245 

7.256 

7.268 

7.279 

7.290 

4 

7.301 

7.313 

7.324 

7.335 

7.347 

7.358 

7.369 

7.381 

7.392 

7.403 

1.95 

7.415 

7.426 

7.438 

7.449 

7.461 

7.472 

7.484 

7.495 

7.507 

7.518 

17 

6 

7.530 

7.541 

7.553 

7.564 

7.576 

7.587 

7.599 

7.610 

7.622 

7.634 

7 

7.645 

7.657 

7.669 

7.680 

7.692 

7.704 

7.715 

7.727 

7.739 

7.751 

8 

7.762 

7.774 

7.786 

7.798 

7.810 

7.821 

7.833 

7.845 

7.857 

7.869 

9 

7.881 

7.892 

7.904 

7.916 

7.928 

7.940 

7.952 

7.964 

7.976 

7.988 

=  31.0063          I/T«  =  0.0322515  + 


10 

CUBES  (continued) 


MATHEMATICAL  TABLES 


N 

C 

1 

2 

3 

4 

5 

6 

7 

8 

9 

*S 

<'-3 

2.00 

8.000 

8.012 

8.024 

8.036 

8.048 

8.060 

8.072 

8.084 

8.096 

8.108 

12 

1 

8.';21 

8.133 

8.145 

8.157 

8.169 

8.181 

8.194 

8.206 

8.218 

8.230 

2 

8.242 

8.255 

8.267 

8.279 

8.291 

8.304 

8.316 

8.328 

8.341 

8.353 

3 

8.365 

8.378 

8.390 

8.403 

8.415 

8.427 

8.440 

8.452 

8.465 

8.477 

4 

8.490 

8.502 

8.515 

8.527 

8.540 

8.552 

8.565 

8.577 

8.590 

8.603 

2.05 

8.615 

8.628 

8.640 

8.653 

8.666 

8.678 

8.691 

8.704 

8.716 

8.729 

13 

6 

8.742 

8.755 

8.767 

8.780 

8.793 

8.806 

8.818 

8.831 

8.844 

8.857 

7 

8.870 

8.883 

8.895 

8.908 

8.921 

8.934 

8.947 

8.960 

8.973 

8.986 

8 

8.999 

9.012 

9.025 

9.038 

9.051 

9.064 

9.077 

9.090 

9.103 

9.116 

9 

9.129 

9.142 

9.156 

9.169 

9.182 

9.195 

9.208 

9.221 

9.235 

9.248 

2.10 

9.261 

9.274 

9.287 

9.301 

9.314 

9.327 

9.341 

9.354 

9.367 

9.381 

1 

9.394 

9.407 

9.421 

9.434 

9.447 

9.461 

9.474 

9.488 

9.501 

9.515 

2 

9.528 

9.542 

9.555 

9.569 

9.582 

9.596 

9.609 

9.623 

9.636 

9.650 

14 

3 

9.664 

9.677 

9.691 

9.704 

9.718 

9.732 

9.745 

9.759 

9.773 

9.787 

4 

9.800 

9.814 

9.828 

9.842 

9.855 

9.869 

9.883 

9.897 

9.911 

9.925 

2.15 

9.938 

9.952 

9.966 

9.980 

9.994 

10.008 

14 

2.1 

9.94 

10.08 

10.22 

10.36 

10.50 

14 

2 

10.65 

10.79 

10.94 

11.09 

11.24 

11.39 

11.54 

11.70 

11.85 

12.01 

15 

3 

12.17 

12.33 

12.49 

12.65 

12.81 

12.98 

13.14 

13.31 

13.48 

13.65 

16 

4 

13.82 

14.00 

14.17 

14.35 

14.53 

14.71 

14.89 

15.07 

15.25 

15.44 

18 

2.5 

15.62 

15.81 

16.00 

16.19 

16.39 

16.58 

16.78 

16.97 

17.17 

17.37 

20 

6 

17.58 

17.78 

17.98 

18.19 

18.40 

18.61 

18.82 

19.03 

19.25 

19.47 

21 

7 

19.68 

19.90 

20.12 

20.35 

20.57 

20.80 

21.02 

21.25 

21.48 

21.72 

23 

8 

21.95 

22.19 

22.43 

22.67 

22.91 

23.15 

23.39 

23.64 

23.89 

24.14 

24 

9 

24.39 

24.64 

24.90 

25.15 

25.41 

25.67 

25.93 

26.20 

26.46 

26.73 

26 

3.0 

27.00 

27.27 

27.54 

27.82 

28.09 

28.37 

28.65 

28.93 

29.22 

29.50 

28 

1 

29.79 

30.08 

30.37 

30.66 

30.96 

31.26 

31.55 

31.86 

32.16 

32.46 

30 

2 

32.77 

33.08 

33.39 

33.70 

34.01 

34.33 

34.65 

34.97 

35.29 

35.61 

32 

3 

35.94 

36.26 

36.59 

36.93 

37.26 

37.60 

37.93 

38.27 

38.61 

38.96 

34 

4 

39.30 

39.65 

40.00 

40.35 

40.71 

41.06 

41.42 

41.78 

42.14 

42.51 

36 

3.5 

42.88 

43.24 

43.61 

43.99 

44.36 

44.74 

45.12 

45.50 

45.88 

46.27 

39 

6 

46.66 

47.05 

47.44 

47.83 

48.23 

48.63 

49.03 

49.43 

49.84 

50.24 

40 

7 

50.65 

51.06 

51.48 

51.90 

52.31 

52.73 

53.16 

53.58 

54.01 

54.44 

42 

8 

54.87 

55.31 

55.74 

56.18 

56.62 

57.07 

57.51 

57.96 

58.41 

58.86 

44 

9 

59.32 

59.78 

60.24 

60.70 

61.16 

61.63 

62.10 

62.57 

63.04 

63.52 

47 

4.0 

64.00 

64.48 

64.96 

65.45 

65.94 

66.43 

66.92 

67.42 

67.92 

68.42 

49 

1 

68.92 

69.43 

69.93 

70.44 

70.96 

71.47 

71.99 

72.51 

73.03 

73.56 

52 

2 

74.09 

74.62 

75.15 

75.69 

76.23 

76.77 

77.31 

77.85 

78.40 

78.95 

54 

3 

79.51 

80.06 

80.62 

81.18 

81.75 

82.31 

82.88 

83.45 

84.03 

84.60 

58 

4 

85.18 

85.77 

86.35 

86.94 

87.53 

88.12 

88.72 

89.31 

89.92 

90.52 

59 

4.5 

91.12 

91.73 

92.35 

92.96 

93.58 

94.20 

94.82 

95.44 

96.07 

96.70 

62 

6 

97.34 

97.97 

98.61 

99.25 

99.90 

100.54 

64 

6 

100.5 

101.2 

101.8 

102.5 

103.2 

7 

7 

103.8 

104.5 

105.2 

105.8 

106.5 

107.2 

107.9 

108.5 

109.2 

109.9 

7 

8 

110.6 

111.3 

112.0 

112.7 

113.4 

114.1 

114.8 

115.5 

116.2 

116.9 

7 

9 

117.6 

118.4 

119.1 

119.8 

120.6 

121.3 

122.0 

122.8 

123.5 

124.3 

7 

Explanation  of  Table  of  Cubes  (pp.  8-11). 

This  table  gives  the  value  of  N*  for  values  of  N  from  1  to  10,  correct  to  four  figures. 
(Interpolated  values  may  be  in  error  by  1  in  the  fourth  figure.) 

To  find  the  cube  of  a  number  N  outside  the  range  from  1  to  10,  note  that 
moving  the  decimal  point  one  place  in  column  N  is  equivalent  to  moving  it  three 
places  in  the  body  of  the  table.  For  example: 

(4.852)»  =  H4.2;       (0.4852)»  =  0.1142;       (485.2)3  =  114200000 

This  table  may  also  be  used  inversely,  to  give  cube  roots. 


MATHEMATICAL  TABLES 


11 


CUBES  (continued) 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

11 

5.0 

125.0 

125.8 

126.5 

127.3 

128.0 

128.8 

129.6 

130.3 

131.1 

131.9 

8 

1 

132.7 

133.4 

134.2 

135.0 

135.8 

136.6 

137.4 

138.2 

139.0 

139.8 

2 

140.6 

141.4 

142.2 

143.1 

143.9 

144.7 

145.5 

146.4 

147.2 

148.0 

3 

148.9 

149.7 

150.6 

151.4 

152.3 

153.1 

154.0 

154.9 

155.7 

156.6 

9 

4 

157.5 

158.3 

159.2 

160.1 

161.0 

161.9 

162.8 

163.7 

164.6 

165.5 

5.5 

166.4 

167.3 

168.2 

169.1 

170.0 

171.0 

171.9 

172.8 

173.7 

174.7 

6 

175.6 

176.6 

177.5 

178.5 

179.4 

180.4 

181.3 

182.3 

183.3 

184.2 

10 

7 

185.2 

186.2 

187.1 

188.1 

189.1 

190.1 

191.1 

192.1 

193.1 

194.1 

8 

195.1 

196.1 

197.1 

198.2 

199.2 

200.2 

201.2 

202.3 

203.3 

204.3 

9 

205.4 

206.4 

207.5 

208.5 

209.6 

210.6 

211.7 

212.8 

213.8 

214.9 

6.0 

216.0 

217.1 

218.2 

219.3 

220.3 

221.4 

222.5 

223.6 

224.8 

225.9 

II 

1 

227.0 

228.1 

229.2 

230.3 

231.5 

232.6 

233.7 

234.9 

236.0 

237.2 

2 

238.3 

239.5 

240.6 

241.8 

243.0 

244.1 

245.3 

246.5 

247.7 

248.9 

12 

3 

250.0 

251.2 

252.4 

253.6 

254.8 

256.0 

257.3 

258.5 

259.7 

260.9 

4 

262.1 

263.4 

264.6 

265.8 

267.1 

268.3 

269.6 

270.8 

272.1 

273.4 

6.5 

274.6 

275.9 

277.2 

278.4 

279.7 

281.0 

282.3 

283.6 

284.9 

286.2 

13 

6 

287.5 

288.8 

290.1 

291.4 

292.8 

294.1 

295.4 

296.7 

298.1 

299.4 

7 

300.8 

302.1 

303.5 

304.8 

306.2 

307.5 

308.9 

310.3 

311.7 

313.0 

14 

8 

314.4 

315.8 

317.2 

318.6 

320.0 

321.4 

322.8 

324.2 

325.7 

327.1 

9 

328.5 

329.9 

331.4 

332.8 

334.3 

335.7 

337.2 

338.6 

340.1 

341.5 

7.0 

343.0 

344.5 

345.9 

347.4 

348.9 

350.4 

351.9 

353.4 

354.9 

356.4 

15 

1 

357.9 

359.4 

360.9 

362.5 

364.0 

365.5 

367.1 

368.6 

370.1 

371.7 

2 

373.2 

374.8 

376.4 

377.9 

379.5 

381.1 

382.7 

384.2 

385.8 

387.4 

16 

3 

389.0 

390.6 

392.2 

393.8 

395.4 

397.1 

398.7 

400.3 

401.9 

403.6 

4 

405.2 

406.9 

408.5 

410.2 

411.8 

413.5 

415.2 

416.8 

418.5 

420.2 

17 

7.5 

421.9 

423.6 

425.3 

427.0 

428.7 

430.4 

432.1 

433.8 

435.5 

437.2 

6 

439.0 

440.7 

442.5 

444.2 

445.9 

447.7 

449.5 

451.2 

453.0 

454.8 

18 

7 

456.5 

458.3 

460.1 

461.9 

463.7 

465.5 

467.3 

469.1 

470.9 

472.7 

8 

474.6 

476.4 

478.2 

480.0 

481.9 

483.7 

485.6 

487.4 

489.3 

491.2 

9 

493.0 

494.9 

496.8 

498.7 

500.6 

502.5 

504.4 

506.3 

508.2 

510.1 

19 

8.0 

512.0 

513.9 

515.8 

517.8 

519.7 

521.7 

523.6 

525.6 

527.5 

529.5 

1 

531.4 

533.4 

535.4 

537.4 

539.4 

541.3 

543.3 

545.3 

547.3 

549.4 

20 

2 

551.4 

553.4 

555.4 

557.4 

559.5 

561.5 

563.6 

565.6 

567.7 

569.7 

3 

571.8 

573.9 

575.9 

578.0 

580.1 

582.2 

584.3 

586.4 

588.5 

590.6 

21 

4 

592.7 

594.8 

596.9 

599.1 

601.2 

603.4 

605.5 

607.6 

609.8 

612.0 

8.5 

614.1 

616.3 

618.5 

620.7 

622.8 

625.0 

627.2 

629.4 

631.6 

633.8 

22 

6 

636.1 

638.3 

640.5 

642.7 

645.0 

647.2 

649.5 

651.7 

654.0 

656.2 

7 

658.5 

660.8 

663.1 

665.3 

667.6 

669.9 

672.2 

674.5 

676.8 

679.2 

23 

8 

681.5 

683.8 

686.1 

688.5 

690.8 

693.2 

695.5 

697.9 

700.2 

702.6 

24 

9 

705.0 

707.3 

709.7 

712.1 

714.5 

716.9 

719.3 

721.7 

724.2 

726.6 

9.0 

729.0 

731.4 

733.9 

736.3 

738.8 

741.2 

743.7 

746.1 

748.6 

751.1 

25 

1 

753.6 

756.1 

758.6 

761.0 

763.6 

766.1 

768.6 

771.1 

773.6 

776.2 

2 

778.7 

781.2 

783.8 

786.3 

788.9 

791.5 

794.0 

796.6 

799.2 

801.8 

26 

3 

804.4 

807.0 

809.6 

812.2 

814.8 

817.4 

820.0 

822.7 

825.3 

827.9 

4 

830.6 

833.2 

835.9 

838.6 

841.2 

843.9 

846.6 

849.3 

852.0 

854.7 

27 

9.5 

857.4 

860.1 

862.8 

865.5 

868.3 

871.0 

873.7 

876.5 

879.2 

882.0 

6 

884.7 

887.5 

890.3 

893.1 

895.8 

898.6 

901.4 

904.2 

907.0 

909.9 

28 

7 

912.7 

915.5 

918.3 

921.2 

924.0 

926.9 

929.7 

932.6 

935.4 

938.3 

8 

941.2 

944.1 

947.0 

949.9 

952.8 

955.7 

958.6 

961.5 

964.4 

967.4 

29 

9 

970.3 

973.2 

976.2 

979.1 

982.1 

985.1 

988.0 

991.0 

994.0 

997.0 

10.0 

1000.0 

=  3 1.0063 


=  0.0322515  + 


Moving  the  decimal  point  ONE  place  in  N  requires  moving  it  THREE  places  in  body  of 
table  (see  p.  10). 


12  MATHEMATICAL  TABLES 

SQUARE  ROOTS  OF  NUMBERS 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

<*)ttt 
& 

1.0 

1.000 

1.005 

1.010 

1.015 

1.020 

1.025 

1.030 

1.034 

1.039 

1.044 

5 

1 

1.049 

1.054 

1.058 

1.063 

1.068 

1.072 

1.077 

1.082 

1.086 

1.091 

2 

1.095 

1.100 

1.105 

1.109 

1.114 

1.118 

1.122 

1.127 

1.131 

1.136 

4 

3 

1.140 

1.145 

1.149 

1.153 

1.158 

1.162 

1.166 

1.170 

1.175 

1.179 

4 

1.183 

1.187 

1.192 

1.196 

1.200 

1.204 

1.208 

1.212 

1.217 

1.221 

1.5 

1.225 

1.229 

1.233 

1.237 

1.241 

1.245 

1.249 

1.253 

1.257 

1.261 

6 

1.265 

1.269 

1.273 

1.277 

1.281 

1.285 

1.288 

1.292 

1.296 

1.300 

7 

1.304 

1.308 

1.311 

1.315 

1.319 

1.323 

1.327 

1.330 

1.334 

1.338 

8 

1.342 

1.345 

1.349 

1.353 

1.356 

1.360 

1.364 

1.367 

1.371 

1.375 

9 

1.378 

1.382 

1.386 

1.389 

1.393 

1.396 

1.400 

1.404 

1.407 

1.411 

2.0 

1.414 

1.418 

1.421 

1.425 

1.428 

1.432 

1.435 

1.439 

1.442 

1.446 

1 

1.449 

1.453 

1.456 

1.459 

1.463 

»     1.466 

1.470 

1.473 

1.476 

1.480 

3 

2 

1.483 

1.487 

1.490 

1.493 

1.497 

1.500 

1.503 

1.507 

1.510 

1.513 

3 

1.517 

1.520 

1.523 

1.526 

1.530 

1.533 

1.536 

1.539 

1.543 

1.546 

4 

1.549 

1.552 

1.556 

1.559 

1.562 

1.565 

1.568 

1.572 

1.575 

1.578 

2.5 

1.581 

1.584 

1.587 

1.591 

1.594 

1.597 

1.600 

1.603 

1.606 

1.609 

6 

1.612 

1.616 

1.619 

1.622 

1.625 

1.628 

1.631 

1.634 

1.637 

1.640 

7 

1.643 

1.646 

1.649 

1.652 

1.655 

1.658 

1.661 

1.664 

1.667 

1.670 

8 

1.673 

1.676 

1.679 

1.682 

1.685 

1.688 

1.691 

1.694 

1.697 

1.700 

9 

1.703 

1.706 

1.709 

1.712 

1.715 

1.718 

1.720 

1.723 

1.726 

1.729 

3.0 

1.732 

1.735 

1.738 

1.741 

1.744 

1.746 

1.749 

1.752 

1.755 

1.758 

1 

1.761 

1.764 

1.766 

1.769 

1.772 

1.775 

1.778 

1.780 

1.783 

1.786 

2 

1.789 

1.792 

1.794 

1.797 

1.800 

1.803 

1.806 

1.808 

1.811 

1.814 

3 

1.817 

1.819 

1.822 

1.825 

1.828 

1.830 

1.833 

1.836 

1.838 

1.841 

4 

1.844 

1.847 

1.849 

1.852 

1.855 

1.857 

1.860 

1.863 

1.865 

1.868 

3.5 

1.871 

1.873 

1.876 

1.879 

1.881 

1.884 

1.887 

1.889 

1.892 

1.895 

6 

1.897 

1.900 

1.903 

1.905 

1.908 

1.910 

1.913 

1.916 

1.918 

1.921 

7 

1.924 

1.926 

1.929 

1.931 

1.934 

1.936 

1.939 

1.942 

1.944 

1.947 

8 

1.949 

1.952 

1.954 

1.957 

1.960 

1.962 

1.965 

1.967 

1.970 

1.972 

9 

1.975 

1.977 

1.980 

1.982 

1.985 

1.987 

1.990 

1.992 

1.995 

1.997 

4.0 

2.000 

2.002 

2.005 

2.007 

2.010 

2.012 

2.015 

2.017 

2.020 

2.022 

1 

2.025 

2.027 

2.030 

2.032 

2.035 

2.037 

2.040 

2.042 

2.045 

2.047 

2 

2 

2.049 

2.052 

2.054 

2.057 

2.059 

2.062 

2.064 

2.066 

2.069 

2.071 

3 

2.074 

2.076 

2.078 

2.081 

2.083 

2.086 

2.088 

2.090 

2.093 

2.095 

4 

2.098 

2.100 

2.102 

2.105 

2.107 

2.110 

2.112 

2.114 

2.117 

2.119 

4.5 

2.121 

2.124 

2.126 

2.128 

2.131 

2.133 

2.135 

2.138 

2.140 

2.142 

6 

2.145 

2.147 

2.149 

2.152 

2.154 

2.156 

2.159 

2.161 

2.163 

2.166 

7 

2.168 

2.170 

2.173 

2.175 

2.177 

2.179 

2.182 

2.184 

2.186 

2.189 

8 

2.191 

2.193 

2.195 

2.198 

2.200 

2.202 

2.205 

2.207 

2.209 

2.211 

9 

2.214 

2.216 

2.218 

2.220 

2.223 

2.225 

2.227 

2.229 

2.232 

2.234 

yV=  1.77245  + 


=  0.56419 


1.25331 


1.64872 


Explanation  of  Table  of  Square  Roots  (pp.  12-15). 

This  table  gives  the  values  of  \/N  for  values  of  N  from  1  to  100,  correct  to  four  figures. 
(Interpolated  values  may  be  in  error  by  1  in  the  fourth  figure.) 

To  find  the  square  root  of  a  number  N  outside  the  range  from  1  to  100,  divide 
the  digits  of  the  number  into  blocks  of  two  (beginning  with  the  decimal  point),  and  note 
that  moving  the  decimal  point  two  places  in  N  is  equivalent  to  moving  it  one  place  in 
the  square  root  of  N.  For  example: 

X/2.718  =  1.648;     A/271.  8  -  16.48;     V0.0002718  =  0.01648; 
V27.18  =>  5.213;     -N/2718    -  52.13;     V/oTo02718     -  0.05213. 


MATHEMATICAL  TABLES 
SQUARE  ROOTS  (continued) 


N 

0 

1 

2 

3 

4 

5 

<* 

5.0 

2.236 

2.238 

2.241 

2.243 

2.245 

2.247 

2.249 

2.252 

2.254 

2.256 

2 

1 

2.258 

2.261 

2.263 

2.265 

2.267 

2.269 

2.272 

2.274 

2.276 

2.278 

2 

2.280 

2.283 

2.285 

2.287 

2.289 

2.291 

2.293 

2.296 

2.298 

2.300 

3 

2.302 

2.304 

2.307 

2.309 

2.311 

2.313 

2.315 

2.317 

2.319 

2.322 

4 

2.324 

2.326 

2.328 

2.330 

2.332 

2.335 

2.337 

2.339 

2.341 

2.343 

6.5 

2.345 

2.347 

2.349 

2.352 

2.354 

2.356 

2.358 

2.360 

2.362 

2.364 

6 

2.366 

2.369 

2.371 

2.373 

2.375 

2.377 

2.379 

2.381 

2.383 

2.385 

7 

2.387 

2.390 

2.392 

2.394 

2.396 

2.398 

2.400 

2.402 

2.404 

2.406 

8 

2.408 

2.410 

2.412 

2.415 

2.417 

2.419 

2.421 

2.423 

2.425 

2.427 

9 

2.429 

2.431 

2.433 

2.435 

2.437 

2.439 

2.441 

2.443 

2.445 

2.447 

6.0 

2.449 

2.452 

2.454 

2.456 

2.458 

2.460 

2.462 

2.464 

2.466 

2.468 

1 

2.470 

2.472 

2.474 

2.476 

2.478 

2.480 

2.482 

2.484 

2.486 

2.488 

2 

2.490 

2.492 

2.494 

2.496 

2.498 

2.500 

2.502 

2.504 

2.506 

2.508 

3 

2.510 

2.512 

2.514 

2.516 

2.518 

2.520 

2.522 

2.524 

2.526 

2.528 

4 

2.530 

2.532 

2.534 

2.536 

2.538 

2.540 

2.542 

2.544 

2.546 

2.548 

6.6 

2.550 

2.551 

2.553 

2.555 

2.557 

2.559 

2.561 

2.563 

2.565 

2.567 

6 

2.569 

2.571 

2.573 

2.575 

2.577 

2.579 

2.581 

2.583 

2.585 

2.587 

7 

2.588 

2.590 

2.592 

2.594 

2.596 

2.598 

2.600 

2.602 

2.604 

2.606 

8 

2.608 

2.610 

2.612 

2.613 

2.615 

2.617 

2.619 

2.621 

2.623 

2.625 

9 

2.627 

2.629 

2.631 

2.632 

2.634 

2.636 

2.638 

2.640 

2.642 

2.644 

7.0 

2.646 

2.648 

2.650 

2.651 

2.653 

2.655 

2.657 

2.659 

2.661 

2.663 

1 

2.665 

2.666 

2.668 

2.670 

2.672 

2.674 

2.676 

2.678 

2.680 

2.681 

2 

2.683 

2.685 

2.687 

2.689 

2.691 

2.693 

2.694 

2.696 

2.698 

2.700 

3 

2.702 

2.704 

2.706 

2.707 

2.709 

2.711 

2.713 

2.715 

2.717 

2.718 

4 

2.720 

2.722 

2.724 

2.726 

2.728 

2.729 

2.731 

2.733 

2.735 

2.737 

7.5 

2.739 

2.740 

2.742 

2.744 

2.746 

2.748 

2.750 

2.751 

2.753 

2.755 

6 

2.757 

2.759 

2.760 

2.762 

2.764 

2.766 

2.768 

2.769 

2.771 

2.773 

7 

2.775 

2.777 

2.778 

2.780 

2.782 

2.784 

2.786 

2.787 

2.789 

2.791 

8 

2.793 

2.795 

2.796 

2.798 

2.800 

2.802 

2.804 

2.805 

2.807 

2.809 

9 

2.811 

2.812 

2.814 

2.816 

2.818 

2.820 

2.821 

2.823 

2.825 

2.827 

8.0 

2.828 

2.830 

2.832 

2.834 

2.835 

2.837 

2.839 

2.841 

2.843 

2.844 

2.846 

2.848 

2.850 

2.851 

2.853 

2.855 

2.857 

2.858 

2.860 

2.862 

2 

2.864 

2.865 

2.867 

2.869 

2.871 

2.872 

2.874 

2.876 

2.877 

2.879 

3 

2.881 

2.883 

2.884 

2.886 

2.888 

2.890 

2.891 

2.893 

2.895 

2.897 

4 

2.898 

2.900 

2.902 

2.903 

2.905 

2.907 

2.909 

2.910 

2.912 

2.914 

8.5 

2.915 

2.917 

2.919 

2.921 

2.922 

2.924 

2.926 

2.927 

2.929 

2.931 

6 

2.933 

2.934 

2.936 

2.938 

2.939 

2.941 

2.943 

2.944 

2.946 

2.948 

7 

2.950 

2.951 

2.953 

2.955 

2.956 

2.958 

2.960 

2.961 

2.963 

2.965 

8 

2.966 

2.968 

2.970 

2.972 

2.973 

2.975 

2.977 

2.978 

2.980 

2.982 

9 

2.983 

2.985 

2.987 

2.988 

2.990 

2.992 

2.993 

2.995 

2.997 

2.998 

9.0 

3.000 

3.002 

3.003 

3.005 

3.007 

3.008 

3.010 

3.012 

3.013 

3.015 

1 

3.017 

3.018 

3.020 

3.022 

3.023 

3.025 

3.027 

3.028 

3.030 

3.032 

2 

3.033 

3.035 

3.036 

3.038 

3.040 

3.041 

3.043 

3.045 

3.046 

3.048 

3 

3.050 

3.051 

3.053 

3.055 

3.056 

3.058 

3.059 

3.061 

3.063 

3.064 

4 

3.066 

3.068 

3.069 

3.071 

3.072 

3.074 

3.076 

3.077 

3.079 

3.081 

9.5 

3.082 

3.084 

3.085 

3.087 

3.089 

3.090 

3.092 

3.094 

3.095 

3.097 

6 

3.098 

3.100 

3.102 

3.103 

3.105 

3.106 

3.108 

3.110 

3.111 

3.113 

7 

3.114 

3.116 

3.118 

3.119 

3.121 

3.122 

3.124 

3.126 

3.127 

3.129 

8 

3.130 

3.132 

3.134 

3J35 

3.137 

3.138 

3.140 

3.142 

3.143 

3.145 

9 

3.146 

3.148 

3.150 

3.151 

3.153 

3.154 

3.156 

3.158 

3.159 

3.161 

Moving  the  decimal  point  TWO  places  in  N  requires  moving  it  ONE  place  in  body 
of  table  (see  p.  12). 


14  MATHEMATICAL  TABLES 

SQUARE  ROOTS  (continued) 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

«d 
$* 

10. 

3.162 

3.178 

3.194 

3.209 

3.225 

3.240 

3.256 

3.271 

3.286 

3.302 

16 

1. 

3.317 

3.332 

3.347 

3.362 

3.376 

3.391 

3.406 

3.421 

3.435 

3.450 

15 

2. 

3.464 

3.479 

3.493 

3.507 

3.521 

3.536 

3.550 

3.564 

3.578 

3.592 

14 

3. 

3.606 

3.619 

3.633 

3.647 

3.661 

3.674 

3.688 

3.701 

3.715 

3.728 

4. 

3.742 

3.755 

3.768 

3.782 

3.795 

3.808 

3.821 

3.834 

3.847 

3.860 

13 

15. 

3.873 

3.886 

3.899 

3.912 

3.924 

3.937 

3.950 

3.962 

3.975 

3.987 

6. 

4.000 

4.012 

4.025 

4.037 

4.050 

4.062 

4.074 

4.087 

4.099 

4.111 

12 

7. 

4.123 

4.135 

4.147 

4.159 

4.171 

4.183 

4.195 

4.207 

4.219 

4.231 

8. 

4.243 

4.254 

4.266 

4.278 

4.290 

4.301 

4.313 

4.324 

4.336 

4.347 

9. 

4.359 

4.370 

4.382 

4.393 

4.405 

4.416 

4.427 

4.438 

4.450 

4.461 

11 

20. 

4.472 

4.483 

4.494 

4.506 

4.517 

4.528 

4.539 

4.550 

4.561 

4.572 

1. 

4.583 

4.593 

4.604 

4.615 

4.626 

4.637 

4.648 

4.658 

4.669 

4.680 

2. 

4.690 

4.701 

4.712 

4.722 

4.733 

4.743 

4.754 

4.764 

4.775 

4.785 

3. 

4.796 

4.806 

4.817 

4.827 

4.837 

4.848 

4.858 

4.868 

4.879 

4.889 

10 

4. 

4.899 

4.909 

4.919 

4.930 

4.940 

4.950 

4.960 

4.970 

4.980 

4.990 

25. 

5.000 

5.010 

5.020 

5.030 

5.040 

5.050 

5.060 

5.070 

5.079 

5.089 

6 

5.099 

5.109 

5.119 

5.128 

5.138 

5.148 

5.158 

5.167 

5.177 

5.187 

7. 

5.196 

5.206 

5.215 

5.225 

5.235 

5.244 

5.254 

5.263 

5.273 

5.282 

8. 

5.292 

5.301 

5.310 

5.320 

5.329 

5.339 

5.348 

5.357 

5.367 

5.376 

9 

9. 

5.385 

5.394 

5.404 

5.413 

5.422 

5.431 

5.441 

5.450 

5.459 

5.468 

30. 

5.477 

5.486 

5.495 

5.505 

5.514 

5.523 

5.532 

5.541 

5.550 

5.559 

5.568 

5.577 

5.586 

5.595 

5.604 

5.612 

5.621 

5.630 

5.639 

5.648 

2! 

5.657 

5.666 

5.675 

5.683 

5.692 

5.701 

5.710 

5.718 

5.727 

5.736 

3. 

5.745 

5.753 

5.762 

5.771 

5.779 

5.788 

5.797 

5.805 

5.814 

5.822 

4. 

5.831 

5.840 

5.848 

5.857 

5.865 

5.874 

5.882 

5.891 

5.899 

5.908 

8 

35. 

5.916 

5.925 

5.933 

5.941 

5.950 

5.958 

5.967 

5.975 

5.983 

5.992 

6. 

6.000 

6.008 

6.017 

6.025 

6.033 

6.042 

6.050 

6.058 

6.066 

6.075 

7. 

6.083 

6.091 

6.099 

6.107 

6.116 

6.124 

6.132 

6.140 

6.148 

6.156 

8. 

6.164 

6.173 

6.181 

6.189 

6.197 

6.205 

6.213 

6.221 

6.229 

6.237 

9. 

6.245 

6.253 

6.261 

6.269 

6.277 

6.285 

6.293 

6.301 

6.309 

6.317 

40. 

6.325 

6.332 

6.340 

6.348 

6.356 

6.364 

6.372 

6.380 

6.387 

6.395 

1. 

6.403 

6.411 

6.419 

6.427 

6.434 

6.442 

6.450 

6.458 

6.465 

6.473 

2. 

6.481 

6.488 

6.496 

6.504 

6.512 

6.519 

6.527 

6.535 

6.542 

6.550 

3. 

6.557 

6.565 

6.573 

6.580 

6.588 

6.595 

6.603 

6.611 

6.618 

6.626 

4. 

6.633 

6.641 

6.648 

6.656 

6.663 

6.671 

6.678 

6.686 

6.693 

6.701 

45. 

6.708 

6.716 

6.723 

6.731 

6.738 

6.745 

6.753 

6.760 

6.768 

6.775 

7 

6. 

6.782 

6.790 

6.797 

6.804 

6.812 

6.819 

6.826 

6.834 

6.841 

6.848 

7. 

6.856 

6.863 

6.870 

6.877 

6.885 

6.892 

6.899 

6.907 

6.914 

6.921 

8. 

6.928 

6.935 

6.943 

6.950 

6.957 

6.964 

6.971 

6.979 

6.986 

6.993 

9. 

7.000 

7.007 

7.014 

7.021 

7.029 

7.036 

7.043 

7.050 

7.057 

7.064 

SQUARE  ROOTS  OF  CERTAIN  FRACTIONS 


N 

VN 

N 

VN 

AT 

VN 

N 

VN 

N 

VN 

N 

VN 

y* 

0.7071 

H 

0.7746 

M 

0.7559 

H 

0.3333 

Hi 

0.6455 

M« 

0.7500 

y& 

0.5774 

% 

0.8944  , 

W 

0.8452 

% 

0.4714 

til 

0.7638 

iM« 

0.8292 

g 

0.8165 

H 

0.4082 

M 

0.9258 

V* 

0.6667 

JM2 

0.9574 

13/f« 

0.9014 

0.5000 

% 

0.9129 

M 

0.3536 

% 

0.7454 

H« 

0.2500 

ls/fe 

0.9682 

94 

0.8660 

W 

0.3780 

** 

0.6124 

7^ 

0.8819 

M« 

0.4330 

to 

0.1768 

% 

0.4472 
0.6325 

M 
H 

0.5345 
0.6547 

H 
H 

0.7906 
0.9354 

% 

0.9428 
0.2887 

Me 
Me 

0.5590 
0.6614 

B 

0.1250 
0.1414 

MATHEMATICAL  TABLES 
SQUARE  ROOTS  (continued) 


N 

0 

1 

2 

a 

4 

5 

6 

7 

8 

9 

wsd 
$* 

50. 

7.071 

7.078 

7.085 

7.092 

7.099 

7.106 

7.113 

7.120 

7.127 

7.134 

7 

1. 

7.141 

7.148 

7.155 

7.162 

7.169 

7.176 

7.183 

7.190 

7.197 

7.204 

2. 

7.211 

7.218 

7.225 

7.232 

7.239 

7.246 

7.253 

7.259 

7.266 

7.273 

3. 

7.280 

7.287 

7.294 

7.301 

7.308 

7.314 

7.321 

7.328 

7.335 

7.342 

4. 

7.348 

7.355 

7.362 

7.369 

7.376 

7.382 

7.389 

7.396 

7.403 

7.409 

55. 

7.416 

7.423 

7.430 

7.436 

7.443 

7.450 

7.457 

7.463 

7.470 

7.477 

6. 

7.483 

7.490 

7.497 

7.503 

7.510 

7.517 

7.523 

7.530 

7.537 

7.543 

7. 

7.550 

7.556 

7.563 

7.570 

7.576 

7.583 

7.589 

7.596 

7.603 

7.609 

8. 

7.616 

7.622 

7.629 

7.635 

7.642 

7.649 

7.655 

7.662 

7.668 

7.675 

9. 

7.681 

7.688 

7.694 

7.701 

7.707 

7.714 

7.720 

7.727 

7.733 

7.740 

6 

60. 

7.746 

7.752 

7.759 

7.765 

7.772 

7.778 

7.785 

7.791 

7.797 

7.804 

1. 

7.810 

7.817 

7.823 

7.829 

7.836 

7.842 

7.849 

7.855 

7.861 

7.868 

2. 

7.874 

7.880 

7.887 

7.893 

7.899 

7.906 

7.912 

7.918 

7.925 

7.931 

3. 

7.937 

7.944 

7.950 

7.956 

7.962 

7.969 

7.975 

7.981 

7.987 

7.994 

4. 

8.000 

8.006 

8.012 

8.019 

8.025 

8.031 

8.037 

8.044 

8.050 

8.056 

65. 

8.062 

8.068 

8.075 

8.081 

8.087 

8.093 

8.099 

8.106 

8.112 

8.118 

6. 

8.124 

8.130 

8.136 

8.142 

8.149 

8.155 

8.161 

8.167 

8.173 

8.179 

7. 

8.185 

8.191 

8.198 

8.204 

8.210 

8.216 

8.222 

8.228 

8.234 

8.240 

8. 

8.246 

8.252 

8.258 

8.264 

8.270 

8.276 

8.283 

8.289 

8.295 

8.301 

9. 

8.307 

8.313 

8.319 

8.325 

8.331 

8.337 

8.343 

8.349 

8.355 

8.361 

70. 

8.367 

8.373 

8.379 

8.385 

8.390 

8.396 

8.402 

8.408 

8.414 

8.420 

1. 

8.426 

8.432 

8.438 

8.444 

8.450 

8.456 

8.462 

8.468 

8.473 

8.479 

2. 

8.485 

8.491 

8.497 

8.503 

8.509 

8.515 

8.521 

8.526 

8.532 

8.538 

3. 

8.544 

8.550 

8.556 

8.562 

8.567 

8.573 

8.579 

8.585 

8.591 

8.597 

4. 

8.602 

8.608 

8.614 

8.620 

8.626 

8.631 

8.637 

8.643 

8.649 

8.654 

75. 

8.660 

8.666 

8.672 

8.678 

8.683 

8.689 

8.695 

8.701 

8.706 

8.712 

6. 

8.718 

8.724 

8.729 

8.735 

8.741 

8.746 

8.752 

8.758 

8.764 

8.769 

7. 

8.775 

8.781 

8.786 

8.792 

8.798 

8.803 

8.809 

8.815 

8.820 

8.826 

8. 

8.832 

8.837 

8.843 

8.849 

8.854 

8.860 

8.866 

8.871 

8.877 

8.883 

9. 

8.888 

8.894 

8.899 

8.905 

8.911 

8.916 

8.922 

8.927 

8.933 

8.939 

80. 

8.944 

8.950 

8.955 

8.961 

8.967 

8.972 

8.978 

8.983 

8.989 

8.994 

1. 

9.000 

9.006 

9.011 

9.017 

9.022 

9.028 

9.033 

9.039 

9.044 

9.050 

2« 

9.055 

9.061 

9.066 

9.072 

9.077 

9.083 

9.088 

9.094 

9.099 

9.105 

5 

3. 

9.110 

9.116 

9.121 

9.127 

9.132 

9.138 

9.143 

9.149 

9.154 

9.160 

4. 

9.165 

9.171 

9.176 

9.182 

9.187 

9.192 

9.198 

9.203 

9.209 

9.214 

85. 

9.220 

9.225 

9.230 

9.236 

9.241 

9.247 

9.252 

9.257 

9.263 

9.268 

6. 

9.274 

9.279 

9.284 

9.290 

9.295 

9.301 

9.306 

9.311 

9.317 

9.322 

7. 

9.327 

9.333 

9.338 

9.343 

9.349 

9.354 

9.359 

9.365 

9.370 

9.375 

8. 

9.381 

9.386 

9.391 

9.397 

9.402 

9.407 

9.413 

9.418 

9.423 

9.429 

9. 

9.434 

9.439 

9.445 

9.450 

9.455 

9.460 

9.466 

9.471 

9.476 

9.482 

90. 

9.487 

9.492 

9.497 

9.503 

9.508 

9.513 

9.518 

9.524 

9.529 

9.534 

1. 

9.539 

9.545 

9.550 

9.555 

9.560 

9.566 

9.571 

9.576 

9.581 

9.586 

2. 

9.592 

9.597 

9.602 

9.607 

9.612 

9.618 

9.623 

9.628 

9.633 

9.638 

3. 

9.644 

9.649 

9.654 

9.659 

9.664 

9.670 

9.675 

9.680 

9.685 

9.690 

4. 

9.695 

9.701 

9.706 

9.711 

9.716 

9.721 

9.726 

9.731 

9.737 

9.742 

95. 

9.747 

9.752 

9.757 

9.762 

9.767 

9.772 

9.778 

9.783 

9.788 

9.793 

6. 

9.798 

9.803 

9.808 

9.813 

9.818 

9.823 

9.829 

9.834 

9.839 

9.844 

7. 

9.849 

9.854 

9.859 

9.864 

9.869 

9.874 

9.879 

9.884 

9.889 

9.894 

8. 

9.899 

9.905 

9.910 

9.915 

9.920 

9.925 

9.930 

9.935 

9.940 

9.945 

9. 

9.950 

9.955 

9.960 

9.965 

9.970 

9.975 

9.980 

9.985 

9.990 

9.995 

=  1.77245+ 


0.56419 


=  1.25331 


=  1.64872 


Moving  the  decimal  point  TWO  places  in  N  requires  moving  it  ONE  place  in  body  of 
table  (seep.  12). 


16  MATHEMATICAL  TABLES 

CUBE  ROOTS  OF  NUMBERS 


TV 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

*£ 

4* 

1.0 

1.000 

1.003 

1.007 

1.010 

1.013 

1.016 

1.020 

1.023 

1.026 

1.029 

3 

1 

1.032 

1.035 

1.038 

1.042 

1.045 

1.048 

1.051 

1.054 

1.057 

1.060 

2 

1.063 

1.066 

1.069 

1.071 

1.074 

1.077 

1.080 

1.083 

1.086 

1.089 

3 

1.091 

.094 

1.097 

.100 

1.102 

.105 

1.108 

.111 

1.113 

1.116 

4 

1.119 

.121 

1.124 

.127 

1.129 

.132 

1.134 

.137 

1.140 

1.142 

1.5 

1.145 

.147 

1.150 

.152 

1.155 

.157 

1.160 

.162 

1.165 

1.167 

2 

6 

1.170 

.172 

1.174 

.177 

1.179 

.182 

1.184 

.186 

1.189 

1.191 

7 

1.193 

.196 

1.198 

.200 

1.203 

.205 

1.207 

.210 

1.212 

1.214 

8 

1.216 

1.219 

1.221 

.223 

1.225 

1.228 

1.230 

.232 

1.234 

1.236 

9 

1.239 

1.241 

1.243 

1.245 

1.247 

1.249 

1.251 

.254 

1.256 

1.258 

2.0 

1.260 

1.262 

1.264 

1.266 

1.268 

1.270 

1.272 

.274 

1.277 

1.279 

1 

1.281 

1.283 

1.285 

1.287 

1.289 

1.291 

1.293 

.295 

1.297 

1.299 

2 

1.301 

.303 

1.305 

1.306 

1.308 

1.310 

1.312 

.314 

1.316 

1.318 

3 

1.320 

1.322 

1.324 

1.326 

1.328 

1.330 

1.331 

.333 

1.335 

1.337 

4 

1.339 

1.341 

1.343 

1.344 

1.346 

1.348 

1.350 

.352 

1.354 

1.355 

2.5 

1.357 

1.359 

1.361 

1.363 

1.364 

1.366 

1.368 

.370 

1.372 

1.373 

6 

1.375 

1.377 

1.379 

1.380 

1.382 

1.384 

1.386 

.387 

1.389 

1.391 

7 

1.392 

.394 

1.396 

1.398 

1.399 

1.401 

1.403 

.404 

1.406 

1.408 

I     8 

1.409 

1.411 

1.413 

1.414 

1.416 

.418 

1.419 

.421 

1.423 

1.424 

9 

1.426 

.428 

1.429 

1.431 

1.433 

1.434 

1.436 

1.437 

1.439 

1.441 

3.0 

1.442 

1.444 

1.445 

1.447 

1.449 

.450 

1.452 

1.453 

1.455 

1.457 

1 

1.458 

1.460 

1.461 

1.463 

1.464 

.466 

1.467 

1.469 

1.471 

1.472 

2 

1.474 

.475 

1.477 

1.478 

1.480 

1.481 

1.483 

1.484 

1.486 

1.487 

3 

1.489 

1.490 

1.492 

1.493 

1.495 

1.496 

1.498 

1.499 

1.501 

1.502 

4 

1.504 

1.505 

1.507 

1.508 

1.510 

1.511 

1.512 

1.514 

1.515 

1.517 

3.5 

1.518 

1.520 

1.521 

1.523 

1.524 

1.525 

1.527 

1.528 

1.530 

1.531 

6 

1.533 

.534 

1.535 

1.537 

1.538 

1.540 

1.541 

1.542 

1.544 

1.545 

1 

7 

1.547 

1.548 

1.549 

1.551 

1.552 

1.554 

1.555 

1.556 

1.558 

1.559 

8 

1.560 

1.562 

1.563 

1.565 

1.566 

.567 

1.569 

1.570 

1.571 

1.573 

9 

1.574 

1.575 

1.577 

1.578 

1.579 

1.581 

1.582 

1.583 

1.585 

1.586 

4.0 

1.587 

1.589 

1.590 

1.591 

1.593 

1.594 

1.595 

1.597 

1.598 

1.599 

1 

1.601 

1.602 

1.603 

1.604 

1.606 

1.607 

1.608 

1.610 

1.611 

1.612 

2 

1.613 

1.615 

1.616 

1.617 

1.619 

1.620 

1.621 

1.622 

1.624 

1.625 

3 

1.626 

1.627 

1.629 

1.630 

1.631 

1.632 

1.634 

1.635 

1.636 

1.637 

4 

1.639 

1.640 

1.641 

1.642 

1.644 

1.645 

1.646 

1.647 

1.649 

1.650 

4.5 

1.651 

1.652 

1.653 

1.655 

1.656 

1.657 

1.658 

1.659 

1.661 

1.662 

6 

1.663 

1.664 

1.666 

1.667 

1.668 

1.669 

1.670 

1.671 

1.673 

1.674 

7 

1.675 

1.676 

1.677 

1.679 

1.680 

1.681 

1.682 

1.683 

1.685 

1.686 

8 

1.687 

1.688 

1.689 

1.690 

1.692 

1.693 

1.694 

1.695 

1.696 

1.697 

9 

1.698 

1.700 

1.701 

1.702 

1.703 

1.704 

1.705 

1.707 

1.708 

1.709 

1.46459          l/V^r~=  0.682784 


Explanation  of  Table  of  Cube  Roots  (pp.  16-21). 

This  table  gives  the  values  of  \/TV  for  all  values  of  TV  from  1  to  1000,  correct  to  four 
figures.  (Interpolated  values  may  be  in  error  by  1  in  the  fourth  figure.) 

To  find  the  cube  root  of  a  number  N  outside  the  range  from  1  to  1000,  divide 
the  digits  of  the  number  into  blocks  of  three  (beginning  with  the  decimal  point),  and 
note  that  moving  the  decimal  point  three  places  in  column  N  is  equivalent  to  moving 
it  one  place  in  the  cube  root  of  TV.  For  example: 

•y^.718  =  1.396;     -y^2718       -  13.96;    -^0.000002718  =  0.01396. 

-^27.18  =  3.007;     -^27180      =  30.07;     -^0.00002718     =  0.03007. 

•^271.8  =  6.477;    -^271800   =  64.77;    -^0.0002718      =  0.06477. 


MATHEMATICAL  TABLES 
CUBE  ROOTS   (continued) 


17 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

£ 

6.0 

1.710 

1.711 

1712 

1.713 

1.715 

1.716 

1.717 

1.718 

1.719 

1.720 

1 

1.721 

1.722 

1.724 

1.725 

1.726 

1.727 

1.728 

1.729 

1.730 

1.731 

2 

1.732 

1.734 

1.735 

1.736 

1.737 

1.738 

1.739 

1.740 

1.741 

1.742 

3 

1.744 

1.745 

1.746 

1.747 

1.748 

1.749 

1.750 

1.751 

1.752 

1.753 

4 

1.754 

1.755 

1.757 

1.758 

1.759 

1.760 

1.761 

1.762 

1.763 

1.764 

5.5 

1.765 

1.766 

1.767 

1.768 

1.769 

1.771 

1.772 

1.773 

1.774 

1.775 

6 

1.776 

1.777 

1.778 

1.779 

1.780 

1.781 

1.782 

1.783 

1.784 

1.785 

7 

1.786 

1.787 

1.788 

1.789 

1.790 

1.792 

1.793 

1.794 

1.795 

1.796 

8 

1.797 

1.798 

1.799 

1.800 

1.801 

1.802 

1.803 

1.804 

1.805 

1.806 

9 

1.807 

1.808 

1.809 

1.810 

1.811 

1.812 

1.813 

1.814 

1.815 

1.816 

6.0 

1.817 

1.818 

1.819 

1.820 

1.821 

1.822 

1.823 

1.824 

1.825 

1.826 

1 

1.827 

1.828 

1.829 

1.830 

1.831 

1.832 

1  833 

1.834 

1.835 

1.836 

2 

1.837 

1.838 

1.839 

1.840 

1.841 

1.842 

1.843 

1.844 

1.845 

1.846 

3 

1.847 

1.848 

1.849 

1.850 

1.851 

1.852 

1.853 

1.854 

1.855 

1.856 

4 

1.857 

1.858 

1.859 

1.860 

1.860 

1.861 

1.862 

1.863 

1.864 

1.865 

6.5 

1.866 

1.867 

1.868 

1.869 

1.870 

1.871 

1.872 

1.873 

1.874 

1.875 

6 

1.876 

1.877 

1.878 

1.879 

1.880 

1.881 

1.881 

1.882 

1.883 

1.884 

7 

1.885 

1.886 

1.887 

1.888 

1.889 

1.890 

1.891 

1.892 

1.893 

1.894 

8 

1.895 

1.895 

1.896 

1.897 

1.898 

1.899 

1.900 

1.901 

1.902 

1.903 

9 

1.904 

1.905 

1.906 

1.907 

1.907 

1.908 

1.909 

1.910 

1.911 

1.912 

7.0 

1.913 

1.914 

1.915 

1.916 

1.917 

1.917 

1.918 

1.919 

1.920 

1.921 

1 

1.922 

1.923 

1.924 

1.925 

1.926 

1.926 

1.927 

1.928 

1.929 

1.930 

2 

1.931 

1.932 

1.933 

1.934 

1.935 

1.935 

1.936 

1.937 

1.938 

1.939 

3 

1.940 

1.941 

1.942 

1.943 

1.943 

1.944 

1.945 

1.946 

1.947 

1.948 

4 

1.949 

1.950 

1.950 

1.951 

1.952 

1.953 

1.954 

1.955 

1.956 

1.957 

7.5 

1.957 

1.958 

1.959 

1.960 

1.961 

1.962 

1.963 

1.964 

1.964 

1.965 

6 

1.966 

1.967 

1.968 

1.969 

1.970 

1.970 

1.971 

1.972 

1.973 

1.974 

7 

1.975 

1.976 

1.976 

1.977 

1.978 

1.979 

1.980 

1.981 

1.981 

1.982 

8 

1  983 

1.984 

1.985 

1.986 

1.987 

1.987 

1.988 

1.989 

1.990 

1.991 

9 

1.992 

1.992 

1.993 

1.994 

1.995 

1.996 

1.997 

1.997 

1.998 

1.999 

8.0 

2.000 

2.001 

2.002 

2.002 

2.003 

2.004 

2.005 

2.006 

2.007 

2.007 

1 

2.008 

2.009 

2.010 

2.01  1 

2.012 

2.012 

2.013 

2.014 

2.015 

2.016 

2 

2.017 

2.017 

2.018 

2.019 

2.020 

2.021 

2.021 

2.022 

2.023 

2.024 

3 

2.025 

2.026 

2.026 

2.027 

2.028 

2.029 

2.030 

2.030 

2.031 

2.032 

4 

2.033 

2.034 

2.034 

2.035 

2.036 

2.037 

2.038 

2.038 

2.039 

2.040 

8.5 

2.041 

2.042 

2.042 

2.043 

2.044 

2.045 

2.046 

2.046 

2.047 

2.048 

6 

2.049 

2.050 

2.050 

2.051 

2.052 

2.053 

2.054 

2.054 

2.055 

2.056 

7 

2.057 

2.057 

2.058 

2.059 

2.060 

2.061 

2.061 

2.062 

2.063 

2.064 

8 

2.065 

2.065 

2.066 

2.067 

2.068 

2.068 

2.069 

2.070 

2.071 

2.072 

9 

2.072 

2.073 

2.074 

2.075 

2.075 

2.076 

2.077 

2.078 

2.079 

2.079 

9.0 

2.080 

2.081 

2.082 

2.082 

2.083 

2.084 

2.085 

2.085 

2.086 

2.087 

1 

2.088 

2.089 

2.089 

2.090 

2.091 

2.092 

2.092 

2.093 

2.094 

2.095 

2 

2.095 

2.096 

2.097 

2.098 

2.098 

2.099 

2.100 

2.101 

2.101 

2.102 

3 

2.103 

2.104 

2.104 

2.105 

2.106 

2.107 

2.107 

2.108 

2.109 

2.110 

4 

2.110 

2.111 

2.112 

2.113 

2.113 

.     2.114 

2.115 

2.116 

2.116 

2.117 

9.5 

2.118 

2.119 

2.119 

2.120 

2.121 

2.122 

2.122 

2.123 

2.124 

2.125 

6 

2.125 

2.126 

2.127 

2.128 

2.128 

2.129 

2.130 

2.130 

2.131 

2.132 

7 

2.133 

2.133 

2.134 

2.135 

2.136 

2.136 

2.137 

2.138 

2.139 

2.139 

8 

2.140 

2.141 

2.141 

2.142 

2.143 

2.144 

2.144 

2.145 

2.146 

2.147 

9 

2.147 

2.148 

2.149 

2.149 

2.150 

2.151 

2.152 

2.152 

2.153 

2.154 

I 

Moving  the  decimal  point  THREE  places  in    N  requires   moving  it   ONE   place  in 
body  of  table  (see  p.  16). 
2 


18 


MATHEMATICAL  TABLES 


CUBE  BOOTS  (continued) 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

^ 
<j-o 

10. 

2.154 

2.162 

2.169 

2.176 

2.183 

2.190 

2.197 

2.204 

2.210 

2.217 

7 

1. 

2.224 

2.231 

2.237 

2.244 

2.251 

2.257 

2.264 

2.270 

2.277 

2.283 

6 

2. 

2.289 

2.296 

2.302 

2.308 

2.315 

2.321 

2.327 

2.333 

2.339 

2.345 

3. 

2.351 

2.357 

2.363 

2.369 

2.375 

2.381 

2.387 

2.393 

2.399 

2.404 

4. 

2.410 

2.416. 

2.422 

2.427 

2.433 

2.438 

2.444 

2.450 

2.455 

2.461 

15. 

2.466 

2.472 

2.477 

2.483 

2.488 

2.493 

2.499 

2.504 

2.509 

2.515 

5 

6. 

2.520 

2.525 

2.530 

2.535 

2.541 

2.546 

2.551 

2.556 

2.561 

2.566 

7. 

2.571 

2.576 

2.581 

2.586 

2.591 

2.596 

2.601 

2.606 

2.611 

2.616 

8. 

2.621 

2.626 

2.630 

2.635 

2.640 

2.645 

2.650 

2.654 

2.659 

2.664 

9. 

2.668 

2.673 

2.678 

2.682 

2.687 

2.692 

2.696 

2.701 

2.705 

2.710 

20. 

2.714 

2.719 

2.723 

2.728 

2.732 

2.737 

2.741 

2.746 

2.750 

2.755 

4 

1. 

2.759 

2.763 

2.768 

2.772 

2.776 

2.781 

2.785 

2.789 

2.794 

2.798 

2. 

2.802 

2.806 

2.811 

2.815 

2.819 

2.823 

2.827 

2.831 

2.836 

2.840 

3. 

2.844 

2.848 

2.852 

2.856 

2.860 

2.864 

2.868 

2.872 

2.876 

2.880 

4. 

2.884 

2.888 

2.892 

2.896 

2.900 

2.904 

2.908 

2.912 

2.916 

2.920 

25. 

2.924 

2.928 

2.932 

2.936 

2.940 

2.943 

2.947 

2.951 

2.955 

2.959 

6. 

2.962 

2.966 

2.970 

2.974 

2.978 

2.981 

2.985 

2.989 

2.993 

2.996 

7. 

3.000 

3.004 

3.007 

3.011 

3.015 

3.018 

3.022 

3.026 

3.029 

3.033 

8. 

3.037 

3.040 

3.044 

3.047 

3.051 

3.055 

3.058 

3.062 

3.065 

3.069 

9. 

3.072 

3.076 

3.079 

3.083 

3.086 

3.090 

3.093 

3.097 

3.100 

3.104 

30. 

3.107 

3.111 

3.114 

3.118 

3.121 

3.124 

3.128 

3.131 

3.135 

3.138 

3 

1. 

3.141 

3.145 

3.148 

3.151 

3.155 

3.158 

3.162 

3.165 

3.168 

3.171 

2. 

3.175 

3.178 

3.181 

3.185 

3.188 

3.191 

3.195 

3.198 

3.201 

3.204 

3. 

3.208 

3.21  1 

3.214 

3.217 

3.220 

3.224 

3.227 

3.230 

3.233 

3.236 

4. 

3.240 

3.243 

3.246 

3.249 

3.252 

3.255 

3.259 

3.262 

3.265 

3.268 

35. 

3.271 

3.274 

3.277 

3.280 

3.283 

3.287 

3.290 

3.293 

3.296 

3.249 

6. 

3.302 

3.305 

3.308 

3.311 

3.314 

3.317 

3.320 

3.323 

3.326 

3.329 

7. 

3.332 

3.335 

3.338 

3.341 

3.344 

3.347 

3.350 

3.353 

3.356 

3.359 

8. 

3.362 

3.365 

3.368 

3.371 

3.374 

3.377 

3.380 

3.382 

3.385 

3.388 

9. 

3.391 

3.394 

3.397 

3.400 

3.403 

3.406 

3.409 

3.411 

3.414 

3.417 

40. 

3.420 

3.423 

3.426 

3.428 

3.431 

3.434 

3.437 

3.440 

3.443 

3.445 

I. 

3.448 

3.451 

3.454 

3.457 

3.459 

3.462 

.3.465 

3.468 

3.471 

3.473 

2. 

3.476 

3.479 

3.482 

3.484 

3.487 

3.490 

3.493 

3.495 

3.498 

3.501 

3. 

3.503 

3.506 

3.509 

3.512 

3.514 

3.517 

3.520 

3.522 

3.525 

3.528 

4. 

3.530 

3.533 

3.536 

3.538 

3.541 

3.544 

3.546 

3.549 

3.552 

3.554 

45. 

3.557 

3.560 

3.562 

3.565 

3.567 

3.570 

3.573 

3.575 

3.578 

3.580 

6. 

3.583 

3.586 

3.588 

3.591 

3.593 

3.596 

3.599 

3.601 

3.604 

3.606 

7. 

3.609 

3.611 

3.614 

3.616 

3.619 

3.622 

3.624 

3.627 

3.629 

3.632 

8. 

3.634 

3.637 

3.639 

3.642 

3.644 

3.647 

3.649 

3.652 

3.654 

3.657 

2 

9. 

3.659 

3.662 

3.664 

3.667 

3.669 

3.672 

3.674 

3.677 

3.679 

3.682 

CUBE  ROOTS  OF  CERTAIN  FRACTIONS 


N 

y* 

N 

& 

2V 

#8 

•  N 

VN 

2V 

j® 

2V 

^2V 

g 

y* 

H 

% 
% 

.7937 
.6934 
.8736 
.6300 
.9086 
.5848 
.7368 

% 
% 
H 
% 
M 
M 
.  y, 

.8434 
.9283 
.5503 
.9410 
.5228 
.6586 
.7539 

¥t 

It 

tt 

H 
W 
Vk 

.8298 
.8939 
.9499 
.5000 
.7211 
.8550 
.9565 

H 
H 

% 
% 
% 
% 
Ha 

.4807 
.6057 
.7631 
.8221 
.9196 
.9615 
.4368 

M2 
M2 
Hl2 

He 
?!  6 
Me 
Me 

.7469 
.8355 
.9714 
.3969 
.5724 
.6786 
.7591 

We 

Hi6 

"/1  6 

*M6 
& 

H4 

Ho 

.8255 
.8826 
.9331 
.9787 
.3150 
.2500 
.2714 

MATHEMATICAL  TABLES 
CUBE  ROOTS  (continued) 


19 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

!i 

60. 

3.684 

3.686 

3.689 

3.691 

3.694 

3.696 

3*99 

3.701 

3.704 

3.706 

2 

1. 

3.708 

3.71  1 

3.713 

3.716 

3.718 

3.721 

3723 

3.725 

3.728 

3.730 

2. 

3.733 

3.735 

3.737 

3.740 

3.742 

3.744 

3.747 

3.749 

3.752 

3.754 

3. 

3.756 

3.759 

3.761 

3.763 

2.766 

3.768 

3.770 

3.773 

3.775 

3.777 

4. 

3.780 

3.782 

3.784 

3.787 

3.789 

3.791 

3.794 

3.796 

3.798 

3.801 

55. 

3.803 

3.805 

3.808 

•  3.810 

3.812 

3.814 

3.817 

.  3.819 

3.821 

3.824 

6. 

3.826 

3.828 

3.830 

3.833 

3.835 

3.837 

3.839 

3.842 

3.844 

3.846 

7. 

3.849 

3.851 

3.853 

3.855 

3.857 

3.860 

3.862 

3.864 

3.866 

3.869 

8. 

3.871 

3.873 

3.875 

3.878 

3.880 

3.882 

3.884 

3.886 

3.889 

3.891 

9. 

3.893 

3.895 

3.897 

3.900 

3.902 

3.904 

3.906 

3.908 

3.911 

3.913 

60. 

3.915 

3.917 

3.919 

3.921 

3.924 

3.926 

3.928 

3.930 

3.932 

3.934 

3.936 

3.939 

3.941 

3.943 

3.945 

3.947 

3.949 

3.951 

3.954 

3.956 

2! 

3.958 

3.960 

3.962 

3.964 

3.966 

3.968 

3.971 

3.973 

3.975 

3.977 

3. 

3.979 

3.981 

3.983 

3.985 

3.987 

3.990 

3.992 

3.994 

3.996 

3.998 

4. 

4.000 

4.002 

4.004 

4.006 

4.008 

4.010 

4.012 

4.015 

4.017 

4.019 

65. 

4.021 

4.023 

4.025 

4.027 

4.029 

4.031 

4.033 

4.035 

4.037 

4.039 

6. 

4.041 

4.043 

4.045 

4.047 

4.049 

4.051 

4.053 

4.055 

4.058 

4.060 

7. 

4.062 

4.064 

4.066 

4.068 

4.070 

4.072 

4.074 

4.076 

4.078 

4.080 

8. 

4.082 

4.084 

4.086 

4.088 

4.090 

4.092 

4.094 

4.096 

4.098 

4.100 

9. 

4.102 

4.104 

4.106 

4.108 

4.109 

4.111 

4.113 

4.115 

4.117 

4.119 

70. 

4.121 

4.123 

4.125 

4.127 

4.129 

4.131 

4.133 

4.135 

4.137 

4.139 

1. 

4.141 

4.143 

4.145 

4.147 

4.149 

4.151 

4.152 

4.154 

4.156 

4.158 

2. 

4.160 

4.162 

4.164 

4.166 

4.168 

4.170 

4.172 

4.174 

4.176 

4.177 

3. 

4.179 

4.181 

4.183 

4.185 

4.187 

4.189 

4.191 

4.193 

4.195 

4.196 

4. 

4.198 

4.200 

4.202 

4.204 

4.206 

4.208 

4.210 

4.212 

4.213 

4.215 

75. 

4.217 

4.219 

4.221 

4.223 

4.225 

4.227 

4.228 

4.230 

4.232 

4.234 

6. 

4.236 

4.238 

4.240 

4.241 

4.243 

4.245 

4.247 

4.249 

4.251 

4.252 

7. 

4.254 

4.256 

4.258 

4.260 

4.262 

4.264 

4.265 

4.267 

4.269 

4.271 

8. 

4.273 

4.274 

4.276 

4.278 

4.280 

4282 

4.284 

4.285 

4.287 

4.289 

9. 

4.291 

4.293 

4294 

4296 

4.298 

4.300 

4.302 

4.303 

4.305 

4.307 

80. 

4.309 

4.311 

4.312 

4.314 

4.316 

4.318 

4.320 

4.321 

4.323 

4.325 

1. 

4.327 

4.329 

4.330 

4.332 

4.334 

4.336 

4.337 

4.339 

4.341 

4.343 

2. 

4.344 

4.346 

4.348 

4.350 

4.352 

4.353 

4.355 

4.357 

4.359 

4.360 

3. 

4.362 

4.364 

4.366 

4.367 

4.369 

4.37  1 

4.373 

4.374 

4.376 

4.378 

4, 

4.380 

4.381 

4.383 

4.385 

4.386 

4.388 

4.390 

4.392 

4.393 

4.395 

85. 

4.397 

4.399 

4.400 

4.402 

4.404 

4.405 

4.407 

4.409 

4.411 

4.412 

6. 

4.414 

4.416 

4.417 

4.419 

4.421 

4.423 

4.424 

4.426 

4.428 

4.429 

7. 

4.431 

4.433 

4.434 

4.436 

4.438 

4.440 

4.441 

4.443 

4.445 

4.446 

8. 

4.448 

4.450 

4.451 

4.453 

4.455 

4.456 

4.458 

4.460 

4.461 

4.463 

9. 

4.465 

4.466 

4.468 

4.470 

4.471 

4.473 

4.475 

4.476 

4.478 

4.480 

90. 

4.481 

4.483 

4.485 

4.486 

4.488 

4.490 

4.491 

4.493 

4.495 

4.496 

1. 

4.498 

4.500 

4.501 

4.503 

4.505 

4.506 

4.508 

4.509 

4.511 

4.513 

2. 

4.514 

4.516 

4.518 

4.519 

4.521 

4.523 

4.524 

4.526 

4.527 

4.529 

3. 

4.531 

4.532 

4.534 

4.536 

4.537 

4.539 

4.540 

4.542 

4.544 

4.545 

4. 

4.547 

4.548 

4.550 

4.552 

4.553 

4.555 

4.556 

4.558 

4.560 

4.561 

95. 

4.563 

4.565 

4.566 

4.568 

4.569 

4.571 

4.572 

4.574 

4.576 

4.577 

6. 

4.579 

4.580 

4.582 

4.584 

4.585 

4.587 

4.588 

4.590 

4.592 

4.593 

7. 

4.595 

4.596 

4.598 

4.599 

4.601 

4.603 

4.604 

4.606 

4.607 

4.609 

8. 

4.610 

4.612 

4.614 

4.615 

4.617 

4.618 

4.620 

4.621 

4.623 

4.625 

9. 

4.626 

4.628 

4.629 

4.631 

4.632 

4.634 

4.635 

4.637 

4.638 

4.640 

Moving  the  decimal  point  THREE  places  in  .W  requires  moving  it  ONE  place  in  body 
of  table  (seep.  16). 


20 


MATHEMATICAL  TABLES 


CUBE  ROOTS   (continued) 


N 

0. 

1. 

2. 

3. 

4. 

5. 

6. 

7. 

8. 

9. 

!§ 

10 

4642 

4.657 

4.672 

4.688 

4.703 

4.718 

4.733 

4.747 

4.762 

4.777 

15 

1 

4791 

4.806 

4.820 

4.835 

4.849 

4.863 

4.877 

4.891 

4.905 

4.919 

14 

2 

4.932 

4.946 

4.960 

4.973 

4.987 

5.000 

5.013 

5.027 

5.040 

5.053 

13 

3 

5.066 

5.079 

5.092 

5.104 

5.117 

5.130 

5.143 

5.155 

5.168 

5.180 

4 

5.192 

5.205 

5.217 

5.229 

5.241 

5.254 

5.266 

5.278 

5.290 

5.301 

12 

15 

5.313 

5.325 

5.337 

'5.348 

5.360 

5.372 

5,383 

5.395 

5.406 

5.418 

6 

5.429 

5.440 

5.451 

5.463 

5.474 

5.485 

5.496 

5.507 

5.518 

5.529 

!1 

7 

5.540 

5.550 

5.561 

5.572 

5.583 

5.593 

5.604 

5.615 

5.625 

5.636 

8 

5.646 

5.657 

5.667 

5.677 

5.688 

5.698 

5.708 

5.718 

5.729 

5.739 

10 

9 

5.749 

5.759 

5.769 

5.779 

5.789 

5.799 

5.809 

5.819 

5.828 

5.838 

20 

5.848 

5.858 

5.867 

5.877 

5.887 

5.896 

5.906 

5.915 

5.925 

5.934 

1 

5.944 

5.953 

5.963 

5.972 

5.981 

5.991 

6.000 

6.009 

6.018 

6.028 

9 

2 

6.037 

6.046 

6.055 

6.064 

6.073 

6.082 

6.091 

6.100 

6.109 

6.118 

3 

6.127 

6.136 

6.145 

6.153 

6.162 

6.171 

6.180 

6.188 

6.197 

6.206 

4 

6.214 

6.223 

6.232 

6.240 

6.249 

6.257 

6.266 

6.274 

6.283 

6.291 

25 

6.300 

6.308 

6.316 

6.325 

6.333 

6.341 

6.350 

6.358 

6.366 

6.374 

8 

6 

6.383 

6.391 

6.399 

6.407 

6.415 

6.423 

6.431 

6.439 

6.447 

6.455 

7 

6.463 

6.471 

6.479 

6.487 

6.495 

6.503 

6.511 

6.519 

6.527 

6.534 

8 

6.542 

6.550 

6.558 

6.565 

6.573 

6.581 

6.589 

6.596 

6.604 

6.611 

9 

6.619 

6.627 

6.634 

6.642 

6.649 

6.657 

6.664 

6.672 

6.679 

6.687 

30 

6.694 

6.702 

6.709 

6.717 

6.724 

6.731 

6.739 

6.746 

6.753 

6.761 

7 

1 

6.768 

6.775 

6.782 

6.790 

6.797 

6.804 

6.811 

6.818 

6.826 

6.833 

2 

6.840 

6.847 

6.854 

6.861 

6.868 

6.875 

6.882 

6.889 

6.896 

6.903 

3 

6.910 

6.917 

6.924 

6.931 

6.938 

6.945 

6.952 

6.959 

6.966 

6.973 

4 

6.980 

6.986 

6.993 

7.000 

7.007 

7.014 

7.020 

7.027 

7.034 

7.041 

35 

7.047 

7.054 

7.061 

7.067 

7.074 

7.081 

7.087 

7.094 

7.101 

7.107 

6 

7.114 

7.120 

7.127 

7.133 

7.140 

7.147 

7.153 

7.160 

7.166 

7.173 

6 

7 

7.179 

7.186 

7.192 

7.198 

7.205 

7.211 

7.218 

7.224 

7.230 

7.237 

8 

7.243 

7.250 

7.256 

7.262 

7.268 

7.275 

7.281 

7.287 

7.294 

7.300 

9 

7.306 

7.312 

7.319 

7.325 

7.331 

7.337 

7.343 

7.350 

7.356 

7.362 

40 

7.368 

7.374 

7.380 

7.386 

7.393 

7.399 

7.405 

7.411 

7.417 

7.423 

1 

7.429 

7.435 

7.441 

7.447 

7.453 

7.459 

7.465 

7.471 

7.477 

7.483 

2 

7.489 

7.495 

7.501 

7.507 

7.513 

7.518 

7.524 

7.530 

7.536 

7.542 

3 

7.548 

7.554 

7.560 

7.565 

7.571 

7.577 

7.583 

7.589 

7.594 

7.600 

4 

7.606 

7.612 

7.617 

7.623 

7.629 

7.635 

7.640 

7.646 

7.652 

7.657 

45 

7.663 

7.669 

7.674 

7.680 

7.686 

7.691 

7.697 

7.703 

7.708 

7.714 

5 

6 

7.719 

7.725 

7.731 

7.736 

7.742 

7.747 

7.753 

7.758 

7.764 

7.769 

7 

7.775 

7.780 

7.786 

7.791 

7.797 

7.802 

7.808 

7.813 

7.819 

7.824 

8 

7.830 

7.835 

7.841 

7.846 

7.851 

7.857 

7.862 

7.868 

7.873 

7.878 

9 

7.884 

7.889 

7.894 

7.900 

7.905 

7.910 

7.916 

7.921 

7.926 

7.932 

AUXILIARY  TABLE  OP  TWO-THIRDS  POWERS 

AND  THREE-HALVES  POWERS     (see  pp.  22-23) 

(To  assist  in  locating  the  decimal  point) 


N 

NK(  =  ^N^) 

N**(-  VF') 

.0001 
.001 
.01 

.1 
1. 

10. 
100. 

.002154 
.01 
.0464 
.  .2154 
1. 
4.64 
21.54 

.000001 
00003162 
.001 
.03162278 
1. 
31.62278 
1000. 

For     complete     table 
of  three-halves  pow- 
ers,  see    pp.    22-23. 
That  table,  used  in- 
versely,   provides    a 
complete     table     of 
two-thirds  powers. 

1000. 

100. 

31622.78 

10000. 

464.16 

1000000. 

MATHEMATICAL  TABLES 
CUBE  ROOTS  (continued) 


21 


N 

0. 

1. 

2. 

a. 

4. 

5. 

6. 

7. 

8. 

9. 

fts 

<;-a 

50 

7.937 

7.942 

7.948 

7.953 

7.958 

7.963 

7.969 

7.974 

7.979 

7.984 

5 

1 

7.990 

7.995 

8.000 

8.005 

8.010 

8.016 

8.021 

8.026 

8.031 

8.036 

2 

8.041 

8.047 

8.052 

8.057 

8.062 

8.067 

8.072 

8.077 

8.082 

8.088 

3 

8.093 

8.098 

8.103 

8.108 

8.113 

8.118 

8.123 

8.128 

8.133 

8.138 

4 

8.143 

8.148 

8.153 

8.158 

8.163 

8.168 

8.173 

8.178 

8.183 

8.188 

55 

8.193 

8.198 

8.203 

8.208 

8.213 

8.218 

8.223 

8.228 

8.233 

8.238 

6 

8.243 

8.247 

8.252 

8.257 

8.262 

8.267 

8.272 

8.277 

8.282 

8.286 

7 

8.291 

8.296 

8.301 

8.306 

8.311 

8.316 

8.320 

8.325 

8.330 

8.335 

8 

8.340 

8.344 

8.349 

8.354 

8.359 

8.363 

8.368 

8.373 

8.378 

8.382 

9 

8.387 

8.392 

8.397 

8.401 

8.406 

8.411 

8.416 

8.420 

8.425 

8.430 

60 

8.434 

8.439 

8.444 

8.448 

8.453 

8.458 

8.462 

8.467 

8.472 

8.476 

1 

8.481 

8.486 

8.490 

8.495 

8.499 

8.504 

8.509 

8.513 

8.518 

8.522 

2 

8.527 

8.532 

8.536 

8.541 

8.545 

8.550 

8.554 

8.559 

8.564 

8.568 

3 

8.573 

8.577 

8.582 

8.586 

8.591 

8.595 

8.600 

8.604 

8.609 

8.613 

4 

4 

8.618 

8.622 

8.627 

8.631 

8.636 

8.640 

8.645 

8.649 

8.653 

8.658 

65 

8.662 

8.667 

8.671 

8.676 

8.680 

8.685 

8.689 

8.693 

8.698 

8.702 

6 

8.707 

8.711 

8.715 

8.720 

8.724 

8.729 

8.733 

8.737 

8.742 

8.746 

7 

8.750 

8.755 

8.759 

8.763 

8.768 

8.772 

8.776 

8.781 

8.785 

8.789 

8 

8.794 

8.798 

8.802 

8.807 

8.811 

8.815 

8.819 

8.824 

8.828 

8.832 

9 

8.837 

8.841 

8.845 

8.849 

8.854 

8.858 

8.862 

8.866 

8.871 

8.875 

70 

8.879 

8.883 

8.887 

8.892 

8.896 

8.900 

8.904 

8.909 

8.913 

8.917 

1 

8.921 

8.925 

8.929 

8.934 

8.938 

8.942 

8.946 

8.950 

8.955 

8.959 

2 

8.963 

8.967 

8.971 

8.975 

8.979 

8.984 

8.988 

8.992 

8.996 

9.000 

3 

9.004 

9.008 

9.012 

9.016 

9.021 

9.025 

9.029 

9.033 

9.037 

9.041 

4 

9.045 

9.049 

9.053 

9.057 

9.061 

9.065 

9.069 

9.073 

9.078 

9.082 

75 

9.086 

9.090 

9.094 

9.098 

9.102 

9.106 

9.110 

9.114 

9.118 

9.122 

6 

9.126 

9.130 

9.134 

9.138 

9.142 

9.146 

9.150 

9.154 

9.158 

9.162 

7 

9.166 

9.170 

9.174 

9.178 

9.182 

9.185 

9.189 

9.193 

9.197 

9.201 

8 

9.205 

9.209 

9.213 

9.217 

9.221 

9.225 

9.229 

9.233 

9.237 

9.240 

9 

9.244 

9.248 

9.252 

9.256 

9.260 

9.264 

9.268 

9.272 

9.275 

9.279 

80 

9.283 

9.287 

9.291 

9.295 

9.299 

9.302 

9.306 

9.310 

9.314 

9.318 

1 

9.322 

9.326 

9.329 

9.333 

9.337 

9.341 

9.345 

9.348 

9.352 

9.356 

2 

9.360 

9.364 

9.368 

9.371 

9.375 

9.379 

9.383 

9.386 

9.390 

9.394 

3 

9.398 

9.402 

9.405 

9.409 

9.413 

9.417 

9.420 

9.424 

9.428 

9.432 

4 

9.435 

9.439 

9.443 

9.447 

9.450 

9.454 

9.458 

9.462 

9.465 

9.469 

85 

9.473 

9.476 

9.480 

9.484 

9.488 

9.491 

9.495 

9.499 

9.502 

9.506 

6 

9.510 

9.513 

9.517 

9.521 

9.524 

9.528 

9.532 

9.535 

9.539 

9.543 

7 

9.546 

9.550 

9.554 

9.557 

9.561 

9.565 

9.568 

9.572 

9.576 

9.579 

8 

9.583 

9.586 

9.590 

9.594 

9.597 

9.601 

9.605 

9.608 

9.612 

9.615 

9 

9.619 

9.623 

9.626 

9.630 

9.633 

9.637 

9.641 

9.644 

9.648 

9.651 

90 

9.655 

9.658 

9.662 

9.666 

9.669 

9.673 

9.676 

9.680 

9.683 

9.687 

1 

9.691 

9.694 

9.698 

9.701 

9.705 

9.708 

9.712 

9.715 

9.719 

9.722 

2 

9.726 

9.729 

9.733 

9.736 

9.740 

9.743 

9.747 

9.750 

9.754 

9.758 

3 

9.761 

9.764 

9.768 

9.771 

9.775 

9.778 

9.782 

9.785 

9.789 

9.792 

4 

9.796 

9.799 

9.803 

9.806 

9.810 

9.813 

9.817 

9.820 

9.824 

9.827 

95 

9.830 

9.834 

9.837 

9.841 

9.844 

9.848 

9.851 

9.855 

9.858 

9.861 

6 

9.865 

9.868 

9.872 

9.875 

9.879 

9.882 

9.885 

9.889 

9.892 

9.896 

7 

9.899 

9.902 

9.906 

9.909 

9.913 

9.916 

9.919 

9.923 

9.926 

9.930 

8 

9.933 

9.936 

9.940 

9.943 

9.946 

9.950 

9.953 

9.956 

9.960 

9.963 

9 

9.967 

9.970 

9.973 

9.977 

9.980 

9.983 

9.987 

9.990 

9.993 

9.997 

100 

10.00 

Moving  the  decimal  point  THREE  places  in  N  requires  moving  it  ONE  place  in  body 
of  table  (see  p.  16). 


22 


MATHEMATICAL  TABLES 


THREE-HALVES  POWERS  OP  NUMBERS     (see  also  p.  20) 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

fl 

1. 

1.000 

1.154 

1.315 

1.482 

1.657 

1.837 

2.024 

2.217 

2.415 

2.619 

183 

2. 

2.828 

3.043 

3.263 

3.488 

3.718 

3.953 

4.192 

4.437 

4.685 

4.939 

237 

3. 

5.196 

5.458 

5.724 

5.995 

6.269 

6.548 

6.831 

7.117 

7.408 

7.702 

280 

4. 

8.000 

8.302 

8.607 

8.917 

9.230 

9.546 

9.866 

10.190 

313 

4. 

10.19 

10.52 

10.85 

33 

5. 

11.18 

11.52 

11.86 

12.20 

12.55 

12.90 

13.25 

13.61 

13.97 

14.33 

35 

6. 

14.70 

15.07 

15.44 

15.81 

16.19 

16.57 

16.96 

17.34 

17.73 

18.12 

38 

7. 

18.52 

18.92 

19.32 

19.72 

20.13 

20.54 

20.95 

21.37 

21.78 

22.20 

41 

8. 

22.63 

23.05 

23.48 

23.91 

24.35 

24.78 

25.22 

25.66 

26.11 

26.55 

44 

9. 

27.00 

27.45 

27.90 

28.36 

28.82 

29.28 

29.74 

30.21 

30.68 

31.15 

46 

10. 

31.62 

32.10 

32.58 

33.06 

33.54 

34.02 

34.51 

35.00 

35.49 

35.99 

49 

1. 

36.48 

36.98 

37.48 

37.99 

38.49 

39.00 

39.51 

40.02 

40.53 

41.05 

51 

2. 

41.57 

42.09 

42.61 

43.14 

43.66 

44.19 

44.73 

45.26 

45.79 

46.33 

53 

3. 

46.87 

47.41 

47.96 

48.50 

49.05 

49.60 

50.15 

50.71 

51.26 

51.82 

55 

4. 

52.38 

52.95 

53.51 

54.08 

54.64 

55.21 

55.79 

56.36 

56.94 

57.51 

57 

15. 

58.09 

58.68 

59.26 

59.85 

60.43 

61.02 

61.62 

62.21 

62.80 

63.40 

59 

6. 

64.00 

64.60 

65.20 

65.81 

66.41 

67.02 

67.63 

68.25 

68.86 

69.48 

61 

7. 

70.09 

70.71 

71.33 

71.96 

72.58 

73.21 

73.84 

74.47 

75.10 

75.73 

63 

8. 

76.37 

77.00 

77.64 

78.28 

78.93 

79.57 

80.22 

80.87 

81.51 

82.17 

65 

9. 

82.82 

83.47 

84.13 

84.79 

85.45 

86.11 

86.77 

87.44 

88.10 

88.77 

66 

20. 

89.44 

90.11 

90.79 

91.46 

92.14 

92.82 

93.50 

94.18 

94.86 

95.55 

68 

1. 

96.23 

96.92 

97.61 

98.30 

99.00 

99.69 

100.38 

69 

I. 

100.4 

101.1 

101.8 

102.5 

7 

2. 

103.2 

103.9 

104.6 

105.3 

106.0 

106.7 

107.4 

108.2 

108.9 

109.6 

7 

3. 

110.3 

111.0 

111.7 

112.5 

113.2 

113.9 

114.6 

115.4 

116.1 

116.8 

7 

4. 

117.6 

118.3 

119.0 

119.8 

120.5 

121.3 

122.0 

122.8 

123.5 

124.3 

7 

25. 

125.0 

125.8 

126.5 

127.3 

128.0 

128.8 

129.5 

130.3 

131.0 

131.8 

8 

6. 

132.6 

133.3 

134.1 

134.9 

135.6 

136.4 

137.2 

138.0 

138.7 

139.5 

8 

7. 

140.3 

141.1 

141.9 

142.6 

143.4 

144.2 

145.0 

145.8 

146.6 

147.4 

8 

8. 

148.2 

149.0 

149.8 

150.5 

151.3 

152.1 

152.9 

153.8 

154.6 

155.4 

8 

9. 

156.2 

157.0 

157.8 

158.6 

159.4 

160.2 

161.0 

161.9 

162.7 

163.5 

8 

30. 

164.3 

"  165.1 

166.0 

166.8 

167.6 

168.4 

169.3 

170.1 

170.9 

171.8 

8 

I, 

172.6 

173  A 

174.3 

175.1 

176.0 

176.8 

177.6 

178.5 

179.3 

180.2 

8 

2. 

181.0 

181.9 

182.7 

183.6 

184.4 

185.3 

186.1 

187.0 

187.8 

188.7 

9 

3. 

189.6 

190.4 

191.3 

192.2 

193.0 

193.9 

194.8 

195.6 

196.5 

197.4 

9 

4. 

198.3 

199.1 

200.0 

200.9 

201.8 

202.6 

203.5 

204.4 

205.3 

206.2 

9 

35. 

207.1 

208.0 

208.8 

209.7 

210.6 

211.5 

212.4 

213.3 

214.2 

215.1 

9 

6. 

216.0 

216.9 

217.8 

218.7 

219.6 

220.5 

221.4 

222.3 

223.2 

224.2 

9 

7. 

225.1 

226.0 

226.9 

227.8 

228.7 

229.6 

230.6 

231.5 

232.4 

233.3 

9 

8. 

234.2 

235.2 

236.1 

237.0 

238.0 

238.9 

239.8 

240.8 

241.7 

242.6 

9 

9. 

243.6 

244.5 

245.4 

246.4 

247.3 

248.3 

249.2 

250.1 

251.1 

252.0 

9 

40. 

253.0 

253.9 

254.9 

255.8 

256.8 

257.7 

258.7 

259.7 

260.6 

261.6 

10 

1; 

262.5 

263.5 

264.5 

265.4 

266.4 

267.3 

268.3 

269.3 

270.2 

271.2 

10 

2. 

272.2 

273.2 

274.1 

275.1 

276.1 

277.1 

278.0 

279.0 

280.0 

281.0 

10 

3. 

282.0 

283.0 

283.9 

284.9 

285.9 

286.9 

287.9 

288.9 

289.9 

290.9 

10 

4. 

291.9 

292.9 

293.9 

294.9 

295.9 

296.9 

297.9 

298.9 

299.9 

300.9 

10 

45. 

301.9 

302.9 

303.9 

304.9 

305.9 

306.9 

307.9 

308.9 

310.0 

311.0 

10 

6. 

312.0 

313.0 

314.0 

315.0 

316.1 

317.1 

318.1 

319.1 

320.2 

321.2 

10 

7. 

322.2 

323.2 

324.3 

325.3 

326.3 

327.4 

328.4 

329.4 

330.5 

331.5 

10 

8. 

332.6 

333.6 

334.6 

335.7 

336.7 

337.8 

338.8 

339.9 

340.9 

342.0 

10 

9. 

343.0 

344.1 

345.1 

346.2 

347.2 

348.3 

349.3 

350.4 

351.4 

352.5 

11 

This  table  gives  N?2  from  N  =>  1  to   N  =  100.     Moving   the   decimal  point  TWO 
places  in  N  requires  moving  it  THREE  places  in  body  of'  table.     Thus: 
(7.23)^  =  19.44;         (723.)^  =  19440;          (0.0723)^  =  0.01944 
(72.3)^  =  614.8;         (7230.)^  =  614800;      (0.723)^  =  0.6148 

Used  inversely,  table  gives  M^  from  M  =  1  to  M  -  1000.    Thus:  (0.6148)**  =>  0.7230. 


MATHEMATICAL   TABLES 
THREE-HALVES  POWERS  (continued')     (See  also  p.  20) 


23 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

£§ 
<H-O 

50. 

353.6 

354.6 

355.7 

356.7 

357.8 

358.9 

359.9 

361.0 

362.1 

363.1 

11 

I. 

364.2 

365.3 

366.4 

367.4 

368.5 

369.6 

370.7 

371.7 

372.8 

373.9 

11 

2. 

375.0 

376.1 

377.1 

378.2 

379.3 

380.4 

381.5 

382.6 

383.7 

384.8 

11 

3 

385.8 

386.9 

388.0 

389.1 

390.2 

391.3 

392.4 

393.5 

394.6 

395.7 

11 

4. 

396.8 

397.9 

399.0 

400.1 

401.2 

402.3 

403.4 

404.6 

405.7 

406.8 

11 

55. 

407.9 

409.0 

410.1 

411.2 

412.3 

413.5 

414.6 

415.7 

416.8 

417.9 

11 

6. 

419.1 

420.2 

421.3 

422.4 

423.6 

424.7 

425.8 

426.9 

428.1 

429.2 

11 

7. 

430.3 

431.5 

432.6 

433.7 

434.9 

436.0 

437.2 

438.3 

439.4 

440.6 

11 

8. 

441.7 

442.9 

444.0 

445.1 

446.3 

447.4 

448.6 

449.7 

450.9 

452.0 

11 

9. 

453.2 

454.3 

455.5 

456.6 

457.8 

459.0 

460.1 

461.3 

462.4 

463.6 

12 

60. 

464.8 

465.9 

467.1 

468.2 

469.4 

470.6 

471.7 

472.9 

474.1 

475.3 

12 

1. 

476.4 

477.6 

478.8 

479.9 

481.1 

482.3 

483.5 

484.6 

485.8 

487.0 

12 

2. 

488.2 

489.4 

490.6 

491.7 

492.9 

'  494.1 

495.3 

496.5 

497.7 

498.9 

12 

3. 

500.0 

501.2 

502.4 

503.6 

504.8 

506.0 

507.2 

508.4 

509.6 

510.8 

12 

4. 

512.0 

513.2 

514.4 

515.6 

516.8 

518.0 

519.2 

520.4 

521.6 

522.8 

12 

65. 

524.0 

525.3 

526.5 

527.7 

528.9 

530.1 

531.3 

532.5 

533.8 

535.0 

12 

6' 

536.2 

537.4 

538.6 

539.8 

541.1 

542.3 

543.5 

544.7 

546.0 

547.2 

12 

7. 

548.4 

.     549.6 

550.9 

552.1 

553.3 

554.6 

555.8 

5570 

558.3 

559.5 

12 

8. 

560.7 

562.0 

563.2 

564.5 

565.7 

566.9 

568.2 

569.4 

570.7 

571.9 

12 

9. 

573.2 

574.4 

575.7 

576.9 

578.1 

579.4 

580.6 

581.9 

583.2 

584.4 

13 

70. 

585.7 

586.9 

588.2 

589.4 

590.7 

591.9 

593.2 

594.5 

595.7 

597.0 

13 

1. 

598.3 

599.5 

600.8 

602.1 

603.3 

604.6 

605.9 

607.1 

608.4 

609.7 

13 

2. 

610.9 

612.2 

613.5 

614.8 

616.0 

617.3 

618.6 

619.9 

621.2 

622.4 

13 

3. 

623.7 

625.0 

626.3 

627.6 

628.8 

630.1 

631.4 

632.7 

634.0 

635.3 

13 

4. 

636.6 

637.9 

639.2 

640.4 

641.7 

643.0 

644.3 

645.6 

646.9 

648.2 

13 

75. 

649.5 

650.8 

652.1 

653.4 

654.7 

656.0 

657.3 

658.6 

659.9 

661.2 

13 

6. 

662.6 

663.9 

665.2 

666.5 

667.8 

669.1 

670.4 

671.7 

673.0 

674.4 

13 

7. 

675.7 

677.0 

678.3 

679.6 

680.9 

682.3 

683.6 

684.9 

686.2 

687.6 

13 

8. 

688.9 

690.2 

691.5 

692.9 

694.2 

695.5 

696.8 

698.2 

699.5 

700.8 

13 

9. 

702.2 

703.5 

704.8 

706.2 

707.5 

708.8 

710.2 

711.5 

712.9 

714.2 

13 

80. 

715.5 

716.9 

718.2 

719.6 

720.9 

722.3 

723.6 

725.0 

726.3 

727.7 

13 

I. 

729.0 

730.4 

731.7 

733.1 

734.4 

735.8 

737.1 

738.5 

739.8 

741.2 

14 

2. 

742.5 

743.9 

745.3 

746.6 

748.0 

749.3 

750.7 

752.1 

753.4 

754.8 

14 

3. 

756.2 

757.5 

758.9 

760.3 

761.6 

763.0 

764.4 

765.8 

767.1 

768.5 

14 

4. 

769.9 

771.2 

772.6 

774.0 

775.4 

776.8 

778.1 

779.5 

780.9 

782.3 

14 

85. 

783.7 

785.0 

786.4 

787.8 

789:2 

790.6 

792.0 

793.4 

794.8 

796.1 

14 

6. 

797.5 

798.9 

800.3 

801.7 

803.1 

804.5 

805.9 

807.3 

808.7 

810.1 

14 

'    7. 

811.5 

812.9 

814.3 

815.7 

817.1 

818.5 

819.9 

821.3 

822.7 

824.1 

14 

8. 

825.5 

826.9 

828.3 

829.7 

831.1 

832.6 

834.0 

835.4 

836.8 

838.2 

14 

9. 

839.6 

841.0 

842.5 

843.9 

845.3 

846.7 

848.1 

849.5 

851.0 

852.4 

14 

90. 

853.8 

855.2 

856.7 

858.1 

859.5 

860.9 

862.4 

863.8 

865.2 

866.7 

14 

1. 

868.  J 

869.5 

870.9 

872.4 

873.8 

875.2 

876.7 

878.1 

879.6 

881.0 

14 

2. 

882.4 

883.9 

885.3 

886.8 

888.2 

889.6 

891.1 

892.5 

894.0 

895.4 

14 

3. 

896.9 

898.3 

899.8 

901.2 

902.7 

904.1 

905.6 

907.0 

908.5 

909.9 

15 

4. 

911.4 

912.8 

914.3 

915.7 

917.2 

918.6 

920.1 

921.6 

923.0 

924.5 

15 

95. 

925.9 

927.4 

928.9 

930.3 

931.8 

933.3 

934.7 

936.2 

937.7 

939.1 

15 

6. 

940.6 

942.1 

943.5 

945.0 

946.5 

948.0 

949.4 

950.9 

952.4 

953.9 

15 

7. 

955.3 

956.8 

958.3 

959.8 

961.3 

962.7 

964.2 

965.7 

967.2 

968.7 

15 

8. 

970.2 

971.6 

973.1 

974.6 

976.1 

977.6 

979.1 

980.6 

982.1 

983.5 

15 

9. 

985.0 

986.5 

988.0 

989.5 

991.0 

992.5 

994.0 

995.5 

997.0 

998.5 

15 

100. 

1000.0 

Moving  the  decimal  point  TWO  places  in  AT  requires  moving  it  THREE  places  in  body 
of  table  (see  also  auxiliary  table  on  p.  20). 


24  MATHEMATICAL  TABLES 

RECIPROCALS  OF  NUMBERS    ' 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

ii 

1.00 

.9990 

.9980 

.9970 

.9960 

.9950 

.9940 

.9930 

.9921 

.9911 

-10 

1 

.9901 

.9891 

.9881 

.9872 

.9862 

.9852 

.9843 

.9833 

.9823 

.9814 

2 

.9804 

.9794 

.9785 

.9775 

.9766 

.9756 

.9747 

.9737  - 

.9728 

.9718 

3 

.9709 

.9699 

.9690 

.9681 

.9671 

.9662 

.9653 

.9643 

.9634 

.9625 

-9 

4 

.9615 

.9606 

.9597 

.9588 

.9579 

.9569 

.9560 

.9551 

.9542 

.9533 

1.05 

.9524 

.9515 

.9506 

.9497 

.9488 

.9479 

.9470 

.9461 

.9452 

.9443 

6 

.9434 

.9425 

.9416 

.9407 

.9398 

.9390 

.9381 

.9372 

.9363 

.9355 

7 

.9346 

.9337 

.9328 

.9320 

.9311 

.9302 

.9294 

.9285 

.9276 

.9268 

8 

.9259 

.9251 

.9242 

.9234 

.9225 

.9217 

.9208 

.9200 

.9191 

.9183 

-8 

9 

.9174 

.9166 

.9158 

.9149 

.9141 

.9132 

.9124 

.9116 

.9107 

.9099 

1.10 

.9091 

.9083 

.9074 

.9066 

.9058 

.9050 

.9042 

.9033 

.9025 

.9017 

.9009 

.9001 

.8993 

.8985 

.8977 

.8969 

.8961 

.8953 

.8945 

.8937 

2 

.8929 

.8921 

.8913 

.8905 

.8897 

.8889 

.8881 

.8873 

.8865 

.8857 

3 

.8850 

.8842 

.8834 

.8826 

.8818 

.8811 

.8803 

.8795 

.8787 

.8780 

4 

.8772 

.8764 

.8757 

.8749 

.8741 

.8734 

.8726 

.8718 

.8711 

.8703 

1.15 

.8696 

.8688 

.8681 

.8673 

.8666 

.8658 

.8651 

.8643 

.8636 

.8628 

6 

.8621 

.8613 

.8606 

.8598 

.8591 

.8584 

.8576 

.8569 

.8562 

.8554 

-7 

7 

.8547 

.8540 

.8532 

.8525 

.8518 

.8511 

.8503 

.8496 

.8489 

.8482 

8 

.8475 

.8467 

.8460 

.8453 

.8446 

.8439 

.8432 

.8425 

.8418 

.8410 

9 

.8403 

.8396 

.8389 

.8382 

.8375 

.8368 

.8361 

.8354 

.8347 

.8340 

1.20 

.8333 

.8326 

.8319 

.8313 

.8306 

.8299 

.8292 

.8285 

.8278 

.8271 

1 

.8264 

.8258 

.8251 

.8244 

.8237 

.8230 

.8224 

.8217 

.8210 

.8203 

2 

.8197 

.8190 

.8183 

.8177 

.8170 

.8163 

.8157 

.8150 

.8143 

.8137 

3 

.8130 

.8123 

.8117 

.8110 

.8104 

.8097 

.8091 

.8084 

.8078 

.8071 

-6 

4 

.8065 

.8058 

.8052 

.8045 

.8039 

.8032 

.8026 

.8019 

.8013 

.8006 

1.25 

.8000 

.7994 

.7987 

.7981 

.7974 

.7968 

.7962 

.7955 

.7949 

.7943 

6 

.7937 

.7930 

.7924 

.7918 

.7911 

.7905 

.7899 

.7893 

.7886 

.7880 

7 

.7874 

.7868 

.7862 

.7855 

.7849 

.7843 

.7837 

.7831 

.7825 

.7819 

8 

.7812 

.7806 

.7800 

.7794 

.7788 

.7782 

.7776 

.7770 

.7764 

.7758 

9 

.7752 

.7746 

.7740 

.7734 

.7728 

.7722 

.7716 

.7710 

.7704 

.7698 

1.30 

.7692 

.7686 

.7680 

.7675 

.7669 

.7663 

.7657 

.7651 

.7645 

.7639 

1 

.7634 

.7628 

.7622 

.7616 

.7610 

.7605 

.7599 

.7593 

.7587 

.7582 

2 

.7576 

.7570 

.7564 

.7559 

.7553 

.7547 

J54J 

.7536 

.7530 

.7524 

3 

.7519 

.7513 

.7508 

.7502 

.7496 

.7491 

.7485 

.7479 

.7474 

.7468 

4 

.7463 

.7457 

.7452 

.7446 

.7440 

.7435 

.7429 

.7424 

.7418 

.7413 

135 

.7407 

.7402 

.7396 

.7391 

.7386 

.7380 

.7375 

.7369 

.7364 

.7358 

-5 

6 

.7353 

.7348 

.7342 

.7337 

.7331 

.7326 

.7321 

.7315 

.7310 

.7305 

7 

.7299 

.7294 

.7289 

.7283 

.7278 

.7273 

.7267 

.7262 

.7257 

.7252 

8 

.7246 

.7241 

.7236 

.7231 

.7225 

.7220 

.7215 

.7210 

.7205 

.7199 

9 

.7194 

.7189 

.7184 

.7179 

.7174 

.7168 

.7163 

.7158 

.7153 

.7148 

1.40 

.7143 

.7138 

.7133 

.7128 

.7123 

.7117 

.7112 

.7107 

.7102 

.7097 

1 

.7092 

.7087 

.7082 

.7077 

.7072 

.7067 

.7062 

.7057 

.7052 

.7047 

2 

.7042 

.7037 

.7032 

.7027 

.7022 

.7018 

.7013 

.7008 

.7003 

.6998 

3 

.6993 

.6988 

.6983 

.6978 

.6974 

.6969 

.6964 

.6959 

.6954 

.6949 

4 

.6944 

.6940 

.6935 

.6930 

.6925 

.6920 

.6916 

.6911 

.6906 

.6901 

1.45 

.6897 

.6892 

.6887 

.6882 

.6878 

.6873 

.6868 

.6863 

.6859 

.6854 

6 

.6849 

.6845 

.6840 

.6835 

.6831 

.6826 

.6821 

.6817 

.6812 

.6807 

7 

.6803 

.6798 

.6793 

.6789 

.6784 

.6780 

.6775 

.6770 

.6766 

.6761 

8 

.6757 

.6752 

.6748 

.6743 

.6739 

.6734 

.6729 

.6725 

.6720 

.6716 

9 

.6711 

.6707 

.6702 

.6698 

.6693 

.6689 

.6684 

.6680 

.6676 

.6671 

1/ir  =  0.318310         1/e  =  0.367879 

Moving  the  decimal  point  in  either  direction  in  N   requires  moving  it  in  the  OPPO- 
SITE direction  in  body  of  table  (see  p.  26). 


MATHEMATICAL  TABLES 
RECIPROCALS   (continued) 


25 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

|1 

1.50 

.6667 

.6662 

.6658 

.6653 

.6649 

.6645 

.6640 

.6636 

.6631 

.6627 

-4 

1 

.6623 

.6618 

.6614 

.6609 

.6605 

.6601 

.6596 

.6592 

.6588 

.6583 

2 

.6579 

.6575 

.6570 

.6566 

.6562 

.6557 

.6553 

.6549 

.6545 

.6540 

3 

.6536 

.6532 

.6527 

.6523 

.6519 

.6515 

.6510 

.6506 

.6502 

.6498 

4 

.6494 

.6489 

.6485 

.6481 

.6477 

.6472 

.6468 

.6464 

.6460 

.6456 

1.55 

.6452 

.6447 

.6443 

.6439 

.6435 

.6431 

.6427 

.6423 

.6418 

.6414 

6 

.6410 

.6406 

.6402 

.6398 

.6394 

.6390 

.6386 

.6382 

.6378 

.6373 

7 

.6369 

.6365 

.6361 

.6357 

.6353 

.6349 

.6345 

.6341 

.6337 

.6333 

8 

.6329 

.6325 

.6321 

.6317 

.6313 

.6309 

.6305 

.6301 

.6297 

.6293 

9 

.6289 

.6285 

.6281 

.6277 

.6274 

.6270 

.6266 

.6262 

.6258 

.6254 

1.60 

.6250 

.6246 

.6242 

.6238 

.6234 

.6231 

.6227 

.6223 

.6219 

.6215 

1 

.6211 

.6207 

.6203 

.6200 

.6196 

.6192 

.6188 

.6184 

.6180 

.6177 

2 

.6173 

.6169 

.6165 

.6161 

.6158 

.6154 

.6150. 

.6146 

.6143 

.6139 

3 

.6135 

.6131 

.6127 

.6124 

.6120 

.6116 

.6112 

.6109 

.6105 

.6101 

4 

.6098 

.6094 

.6090 

.6086 

.6083 

.6079 

.6075 

.6072 

.6068 

.6064 

1.65 

.6061 

.6057 

.6053 

.6050 

.6046 

.6042 

.6039 

.6035 

.6031 

.6028 

6 

.6024 

.6020 

.6017 

.6013 

.6010 

.6006 

.6002 

.5999 

.5995 

.5992 

7 

.5988 

.5984 

.5981 

.5977 

.5974 

.5970 

.5967 

.5963 

.5959 

.5956 

8 

.5952 

.5949 

.5945 

.5942 

.5938 

.5935 

.5931 

.5928 

.5924 

.5921 

9 

.5917 

.5914 

.5910 

.5907 

.5903 

.5900 

.5896 

.5893 

.5889 

.5886 

1.70 

.5882 

.5879 

.5875 

.5872 

.5869 

.5865 

.5862 

.5858 

.5855 

.5851 

-3 

1 

.5848 

.5845 

.5841 

.5838 

.5834 

.5831 

.5828 

.5824 

.5821 

.5817 

2 

.5814 

.5811 

.5807 

.5804 

.5800 

.5797 

.5794 

.5790 

.5787 

.5784 

3 

.5780 

.5777 

.5774 

.5770 

.5767 

.5764 

.5760 

.5757 

.5754 

.5750 

4 

.5747 

.5744 

.5741 

.5737 

.5734 

.5731 

.5727 

.5724 

.5721 

.5718 

1.75 

.5714 

.5711 

.5708 

.5705 

.5701 

.5698 

.5695 

.5692 

.5688 

.5685 

6 

.5682 

.5679 

.5675 

.5672 

.5669 

.5666 

.5663 

.5659 

.5656 

.5653 

7 

.5650 

.5647 

.5643 

.5640 

.5637 

.5634 

.5631 

.5627 

.5624 

.5621 

8 

.5618 

.5615 

.5612 

.5609 

.5605 

.5602 

.5599 

.5596 

.5593 

.5590 

9 

.5587 

.5583 

.5580 

.5577 

.5574 

.5571 

.5568 

.5565 

.5562 

.5559 

1.80 

.5556 

.5552 

.5549 

.5546 

.5543 

.5540 

.5537 

.5534 

.5531 

.5528 

1 

.5525 

.5522 

.5519 

.5516 

.5513 

.5510 

.5507 

.5504 

.5501 

.5498 

2 

.5495 

.5491 

.5488 

.5485 

.5482 

.M79 

.5476 

.5473 

.5470 

.5467 

3 

.5464 

.5461 

.5459 

.5456 

.5453 

.5450 

.5447 

.5444 

.5441 

.5438 

4 

.5435 

.5432 

.5429 

.5426 

.5423 

.5420 

.5417 

.5414 

.5411 

.5408 

1.85 

.5405 

.5402 

.5400 

.5397 

.5394 

.5391 

.5388 

.5385 

.5382 

.5379 

6 

.5376 

.5373 

.5371 

.5368 

.5365 

.5362 

.5359 

.5356 

.5353 

.5350 

7 

.5348 

.5345 

.5342 

.5339 

.5336 

.5333 

.5330 

.5328 

.5325 

.5322 

8 

.5319 

.5316 

.5313 

.5311 

.5308 

.5305 

.5302 

.5299 

.5297 

.5294 

9 

.5291 

.5288 

.5285 

.5283 

.5280 

.5277 

.5274 

.5271 

.5269 

.5266 

1.90 

.5263 

.5260 

.5258 

.5255 

.5252 

.5249 

.5247 

.5244 

.5241 

.5238 

1 

.5236 

.5233 

.5230 

.5227 

.5225 

.5222 

.5219 

.5216 

.5214 

.5211 

2 

.5208 

.5206 

.5203 

.5200 

.5198 

.5195 

.5192 

.5189 

.5187 

.5184 

3 

.5181 

.5179 

.5176 

.5173 

.5171 

.5168 

.5165 

.5163 

.5160 

.5157 

4 

.5155 

.5152 

.5149 

.5147 

.5144 

.5141 

.5139 

.5136 

.5133 

.5131 

1.95 

.5128 

.5126 

.5123 

.5120 

5118 

.5115 

.5112 

.5110 

.5107 

.5105 

6 

.5102 

.5099 

.5097 

.5094 

.5092 

.5089 

.5086 

.5084 

.5081 

.5079 

7 

.5076 

.5074 

.5071 

.5068 

.5066 

.5063 

.5061 

.5058 

.5056 

.5053 

-2 

8 

.5051 

.5048 

.5045 

.5043 

.5040 

.5038 

.5035 

.5033 

.5030 

.5028 

9 

.5025 

.5023 

.5020 

.5018 

.5015 

.5013 

.5010 

.5008 

.5005 

.5003 

Moving  the  decimal  point  in  either  direction  in  N  requires  moving  it  in  the  OPPO- 
SITE direction  in  body  of  table  (see  p.  26). 


26  MATHEMATICAL  TABLES 

RECIPROCALS  (continued) 


N 

o 

j 

2 

3 

. 

2.0 

.5000 

.4975 

.4950 

.4926 

.4902 

.4878 

.4854 

.4831 

.4808 

.4785 

-24 

I 

.4762 

.4739 

.4717 

.4695 

.4673 

.4651 

.4630 

.4608 

.4587 

.4566 

-21 

2 

.4545 

.4525 

.4505 

.4484 

.4464 

.4444 

.4425 

.4405 

.4386 

.4367 

-20 

3 

.4348 

.4329 

.4310 

.4292 

.4274 

.4255 

.4237 

.4219 

.4202 

.4184 

-  18 

4 

.4167 

.4149 

.4132 

.4115 

.4098 

.4082 

.4065 

.4049 

.4032 

.4016 

-  17 

2.5 

.4000 

.3984 

.3968 

.3953 

3937 

.3922 

.3906 

3891 

3876 

386  1 

-  15 

6 

.3846 

.3831 

.3817 

3802 

3788 

3774 

3759 

3745 

3731 

3717 

-  14 

7 

.3704 

.3690 

3676 

.3663 

3650 

.3636 

3623 

3610 

3597 

3584 

-  13 

8 

.3571 

.3559 

.3546 

3534 

.3521 

3509 

3497 

3484 

.3472 

3460 

-  12 

9 

.3448 

.3436 

3425 

3413 

.3401 

3390 

3378 

3367 

.3356 

.3344 

-12 

3.0 

.3333 

.3322 

.3311 

.3300 

3289 

.3279 

3268 

.3257 

3247 

3236 

-  11 

1 

.3226 

.3215 

.3205 

.3195 

.3185 

3175 

.3165 

.3155 

.3145 

3135 

-  10 

2 

.3125 

.3115 

3106 

.3096 

3086 

.3077 

3067 

.3058 

3049 

3040 

-  10 

3 

.3030 

.3021 

.3012 

3003 

.2994 

.2985 

.2976 

.2967 

.2959 

.2950 

-9 

4 

.2941 

.2933 

.2924 

.2915 

.2907 

.2899 

.2890 

.2882 

.2874 

.2865 

-8 

3.5 

.2857 

.2849 

.2841 

.2833 

.2825 

.2817 

.2809 

.2801 

.2793 

.2786 

-8 

6 

.2778 

.2770 

.2762 

.2755 

.2747 

.2740 

.2732 

.2725 

.2717 

.2710 

-8 

7 

.2703 

J2695 

.2688 

.2681 

.2674 

.2667 

.2660 

.2653 

.2646 

.2639 

-7 

8 

.2632 

.2625 

.2618 

.2611 

.2604 

.2597 

.2591 

.2584 

.2577 

.2571 

-7 

9 

.2564 

.2558 

.2551 

.2545 

.2538 

.2532 

.2525 

.2519 

.2513 

.2506 

-6 

4.0 

.2500 

.2494 

.2488 

J2481 

.2475 

.2469 

.2463 

.2457 

.2451 

.2445 

-6 

1 

.2439 

.2433 

.2427 

.2421 

.2415 

.2410 

.2404 

.2398 

.2392 

.2387 

-6 

2 

.2381 

.2375 

.2370 

2364 

.2358 

.2353 

.2347 

.2342 

.2336 

.2331 

-6 

3 

.2326 

.2320 

.2315 

.2309 

.2304 

.2299 

.2294 

.2288 

.2283 

.2278 

-5 

4 

.2273 

.2268 

.2262 

.2257 

.2252 

.2247 

.2242 

.2237 

.2232 

.2227 

-5 

4.5 

.2222 

.2217 

.2212 

.2208 

.2203 

.2198 

.2193 

.2188 

.2183 

.2179 

-5 

6 

.2174 

.2169 

.2165 

.2160 

.2155 

.2151 

.2146 

.2141 

.2137 

.2132 

-5 

7 

.2128 

.2123 

.2119 

2114 

.2110 

.2105 

.2101 

.2096 

.2092 

.2088 

-4 

8 

.2083 

.2079 

.2075 

.2070 

.2066 

.2062 

.2058 

.2053 

.2049 

.2045 

-4 

9 

.2041 

.2037 

.2033 

.2028 

.2024 

.2020 

.2016 

.2012 

.2008 

.2004 

-4 

I/T  =  0.318310       1/e  =  0.367879 


Explanation  of  Table  of  Reciprocals  (pp.  24-27). 

This  table  gives  the  values  of  1/N  for  values  of  N  from  1  to  10,  correct  to  four  figures. 
(Interpolated  values  may  be  in  error  by  1  in  the  fourth  figure.) 

To  find  the  reciprocal  of  a  number  N  outside  the  range  from  1  to  10,  note 
that  moving  the  decimal  point  any  number  of  places  in  either  direction  in  column  N 
is  equivalent  to  moving  it  the  same  number  of  places  in  the  opposite  direction  in  the 
body  of  the  table.  For  example: 

1          0.3108;    -^r  =0.0003108;   nn^n^  -  310.8 


3.217 


3217. 


0.003217 


MATHEMATICAL  TABLES 
RECIPROCALS   (continued) 


27 


I* 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

«d 
& 

5.0 

.2000 

.1996 

.1992 

.1988 

.1984 

.1980 

.1976 

.1972 

.1969 

.1965 

-4 

.1 

.1961 

.1957 

.1953 

.1949 

.1946 

.1942 

.1938 

.1934 

.1931 

.1927 

.2 

.1923 

.1919 

.1916 

.1912 

.1908 

.1905 

.1901 

.1898 

.1894 

.1890 

.3 

.1887 

.1883 

.1880 

.1876 

.1873 

.1869 

.1866 

.1862 

.1859 

.1855 

.4 

.1852 

.1848 

.1845 

.1842 

.1838 

.1835 

.1832 

.1828 

.1825 

.1821 

-3 

5.5 

.1818 

.1815 

.1812 

.1808 

.1805 

.1802 

.1799 

.1795 

.1792 

.1789 

.6 

.1786 

.1783 

.1779 

.1776 

.1773 

.1770 

.1767 

.1764 

.1761 

.1757 

.7 

.1754 

.1751 

.1748 

.1745 

.1742 

.1739 

.1736 

.1733 

.1730 

.1727 

.8 

.1724 

.1721 

.1718 

.1715 

.1712 

.1709 

.1706 

.1704 

.1701 

.1698 

.9 

.1695 

.1692 

.1689 

.1686 

.1684 

.1681 

.1678 

.1675 

.1672 

.1669 

6.0  .1667 

.1664 

.1661 

.1658 

.1656 

.1653 

.1650 

.1647 

.1645 

.1642 

.1 

.1639 

.1637 

.1634 

.1631 

.1629 

.1626 

.1623 

.1621 

.1618 

.1616 

.2 

.1613 

.1610 

.1608 

.1605 

.1603 

.1600 

.1597 

.1595 

.1592 

.1590 

.3 

.1587 

.1585 

.1582 

.1580 

.1577 

.1575 

.1572 

.1570 

.1567 

.1565 

-2 

.4 

.1563 

.1560 

.1558 

.1555 

.1553 

.1550 

.1548 

.1546 

.1543 

.1541 

6.5 

.1538 

.1536 

.1534 

.1531 

.1529 

.1527 

.1524 

.1522 

.1520 

.1517 

.6 

.1515 

.1513 

.1511 

.1508 

.1506 

.1504 

.1502 

.1499 

.1497 

.1495 

.7 

.1493 

.1490 

.1488 

.1486 

.1484 

.1481 

.1479 

.1477 

.1475 

.1473 

.8 

.1471 

.1468 

.1466 

.1464 

.1462 

.1460 

.1458 

.1456 

.1453 

.1451 

.9 

.1449 

.1447 

.1445 

.1443 

.1441 

.1439 

.1437 

.1435 

.1433 

.1431 

7.0 

.1429 

.1427 

.1425 

.1422 

.1420 

.1418 

.1416 

.1414 

.1412 

.1410 

.1408 

.1406 

.1404 

.1403 

.1401 

.1399 

.1397 

.1395 

.1393 

.1391 

'.2 

.1389 

.1387 

.1385 

.1383 

.1381 

.1379 

.1377 

.1376 

.1374 

.1372 

.3 

.1370 

.1368 

.1366 

.1364 

.1362 

.1361 

.1359 

.1357 

.1355 

.1353 

.4 

.1351 

.1350 

.1348 

.1346 

.1344 

.1342 

.1340 

.1339 

.1337 

.1335 

7.5 

.1333 

.1332 

.1330 

.1328 

.1326 

.1325 

.1323 

.1321 

.1319 

.1318 

.6 

.1316 

.1314 

.1312 

.1311 

.1309 

.1307 

.1305 

.1304 

.1302 

.1300 

.7 

.1299 

.1297 

.1295 

.1294 

.1292 

.1290 

.1289 

.1287 

.1285 

.1284 

.8 

.1282 

.1280 

.1279 

.1277 

.1276 

.1274 

.1272 

.1271 

.1269 

.1267 

.9 

.1266 

.1264 

.1263 

.1261 

.1259 

.1258 

.1256 

.1255 

.1253 

.1252 

8.0 

.1250 

.1248 

.1247 

.1245 

.1244 

.1242 

.1241 

.1239 

.1238 

.1236 

.1235 

.1233 

.1232 

.1230 

.1229 

.1227 

.1225 

.1224 

.1222 

.1221 

'.2 

.1220 

.1218 

.1217 

.1215 

.1214 

.1212 

.1211 

.1209 

.1208 

.1206 

.3 

.1205 

.1203 

.1202 

.1200 

.1199 

.1198 

.1196 

.1195 

.1193 

.1192 

.4 

.1190 

.1189 

.1188 

.1186 

.1185 

.1183 

.1182 

.1181 

.1179 

.1178 

-1 

8.5 

.1176 

.1175 

.1174 

.1172 

.1171 

.1170 

.1168 

.1167 

.1166 

.1164 

.6 

.1163 

.1161 

.1160 

.1159 

.1157 

.1156 

.1155 

.1153 

.1152 

.1151 

.7 

.1149 

.1148 

.1147 

.1145 

.1144 

.1143 

.1142 

.1140 

.1139 

.1138 

.8 

.1136 

.1135 

.1134 

.1133 

.1131 

.1130 

.1129 

.1127 

.1126 

.1125 

.9 

.1124 

.1122 

.1121 

.1120 

.1119 

.1117 

.1116 

.1115 

.1114 

.1112 

9.0 

.1111 

.1110 

.1109 

.1107 

.1106 

.1105 

.1104 

.1103 

.1101 

.1100 

.1 

.1099 

.1098 

.1096 

.1095 

.1094 

.1093 

.1092 

.1091 

.1089 

.1088 

.2 

.1087 

.1086 

.1085 

.1083 

.1082 

.1081 

.1080 

.1079 

.1078 

.1076 

.3 

.1075 

.1074 

.1073 

.1072 

.1071 

.1070 

.1068 

.1067 

.1066 

.1065 

.4 

.1064 

.1063 

.1062 

.1060 

.1059 

.1058 

.1057 

.1056 

.1055 

.1054 

9.5 

.1053 

.1052 

.1050 

.1049 

.1048 

.1047 

.1046 

.1045 

.1044 

.1043 

.6 

.1042 

.1041 

.1040 

.1038 

.1037 

.1036 

.1035 

.1034 

.1033 

.1032 

.7 

.1031 

.1030 

.1029 

J028 

.1027 

.1026 

.1025 

.1024 

.1022 

.1021 

.8 

.1020 

.1019 

.1018 

.1017 

.1016 

.1015 

.1014 

.1013 

.1012 

.1011 

.9 

.1010 

.1009 

.1008 

.1007 

.1006 

.1005 

.1004 

.1003 

.1002 

.1001 

Moving  the  decimal  point  in  either  direction  in  N  requires  moving  it  in  the  OPPOSITE 
direction  in  body  of  table  (see  p.  26). 


28 


MATHEMATICAL  TABLES 


CIRCUMFERENCES    OF   CIRCLES   BY   HUNDREDTHS 

(For  circumferences  by  eighths,  see  p.  32) 


D 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

>1 

<T 

1.0 

3.142 

3.173 

3.204 

3.236 

3.267 

3.299 

3.330 

3.362 

3.393 

3.424 

31 

.1 

3.456 

3.487 

3.519 

3.550 

3.581 

3.613 

3.644 

3.676 

3.707 

3.738 

.2 

3.770 

3.801 

3.833 

3.864 

3.896 

3.927 

3.958 

3.990 

4.021 

4.053 

.3 

4.084 

4.115 

4.147 

4.178 

4.210 

4.241 

4.273 

4.304 

4.335 

4.367 

.4 

4.398 

4.430 

4.461 

4.492 

4.524 

4.555 

4.587 

4.618 

4.650 

4.681 

1.5 

4.712 

4.744 

4.775 

4.807 

4.838 

4.869 

4.901 

4.932 

4.964 

4.995 

.6 

5.027 

5.058 

5.089 

5.121 

5.152 

5.184 

5.215 

5.246 

5.278 

5.309 

.7 

5.341 

5.372 

5.404 

5.435 

5.466 

5.498 

5.529 

5.561 

5.592 

5.623 

.8 

5.655 

5.686 

5.718 

5.749 

5.781 

5.812 

5.843 

5.875 

5.906 

5.938 

.9 

5.969 

6.000 

6.032 

6.063 

6.095 

6.126 

6.158 

6.189 

6.220 

6.252 

2.0 

6.283 

6.315 

6.346 

6.377 

6.409 

6.440 

6.472 

6.503 

6.535 

6.566 

.1 

6.597 

6.629 

6.660 

6.692 

6.723 

6.754 

6.786 

6.817 

6.849 

6.880 

.2 

6.912 

6.943 

6.974 

7.006 

7.037 

7.069 

7.100 

7.131 

7.163 

7.194 

.3 

7.226 

7.257 

7.288 

7.320 

7.351 

7.383 

7.414 

7.446 

7.477 

7.508 

.4 

7.540 

7.571 

7.603 

7.634 

7.665 

7.697 

7.728 

7.760 

7.791 

7.823 

2.5 

7.854 

7.885 

7.917 

7.948 

7.980 

8.011 

8.042 

8.074 

8.105 

8.137 

.6 

8.168 

8.200 

8.231 

8.262 

8.294 

8.325 

8.357 

8.388 

8.419 

8.451 

.7 

8.482 

8.514 

8.545 

8.577 

8.608 

8.639 

8.671 

8.702 

8.734 

8.765 

.8 

8.796 

8.828 

8.859 

8.891 

8.922 

8.954 

8.985 

9.016 

9.048 

9.079 

.9 

9.111 

9.142 

9.173 

9.205 

9.236 

9.268 

9.299 

9.331 

9.362 

9.393 

3.0 

9.425 

9.456 

9.488 

9.519 

9.550 

9.582 

9.613 

9.645 

9.676 

9.708 

9.739 

9.770 

9.802 

9.833 

9.865 

9.8% 

9.927 

9.959 

9.990 

10.022 

31 

*1 

10.02 

3 

.2 

10.05 

10.08 

10.12 

10.15 

10.18 

10.21 

10.24 

10.27 

10.30 

10.34 

.3 

10.37 

10.40 

10.43 

10.46 

10.49 

10.52 

10.56 

1059 

10.62 

10.65 

.4 

10.68 

10.71 

10.74 

10.78 

10.81 

10.84 

10.87 

10.90 

10.93 

10.96 

3.5 

11.00 

11.03 

11.06 

11.09 

11.12 

11.15 

11.18 

11.22 

11.25 

11.28 

.6 

11.31 

11.34 

11.37 

11.40 

11.44 

11.47 

11.50 

11.53 

11.56 

11.59 

.7 

11.62 

11.66 

11.69 

11.72 

11.75 

11.78 

11.81 

11.84 

11.88 

11.91 

.8 

11.94 

11.97 

12.00 

12.03 

12.06 

12.10 

12.13 

12.16 

12.19 

12.22 

.9 

12.25 

12.28 

12.32 

12.35 

12.38 

12.41 

12.44 

12.47 

12.50 

12.53 

4.0 

12.57 

12.60 

12.63 

12.66 

12.69 

12.72 

12.75 

12.79 

12.82 

12.85 

12.88 

12.91 

12.94 

12.97 

13.01 

13.04 

13.07 

13.10 

13.13 

13.16 

'.2 

13.19 

13.23 

13.26 

13.29 

13.32 

13.35 

13.38 

13.41 

13.45 

13.48 

.3 

13.51 

13.54 

13.57 

13.60 

13.63 

13.67 

13.70 

13.73 

13.76 

13.79 

.4 

13.82 

13.85 

13.89 

13.92 

13.95 

13.98 

14.01 

14.04 

14.07 

14.11 

4.5 

14.14 

14.17 

14.20 

14.23 

14.26 

14.29 

14.33 

14.36 

14.39 

14.42 

.6 

14.45 

14.48 

14.51 

14.55 

14.58 

14.61 

14.64 

14.67 

14.70 

14.73 

.7 

14.77 

14.80 

14.83 

14.86 

14.89 

14.92 

14.95 

14.99 

15.02 

15.05 

.8 

15.08 

15.11 

15.14 

15.17 

15.21 

15.24 

15.27 

15.30 

15.33 

15.36 

.9 

15.39 

15.43 

15.46 

15.49 

15.52 

15.55 

15.58 

15.61 

15.65 

15.68 

Explanation  of  Table  of  Circumferences  (pp.  28-29) 

This  table  gives  the  product  of  T  times  any  number  D  from  1  to  10;  that  is,  it  is  a  table 
of  multiples  of  IT.     (D  —  diameter.) 

Moving  the  decimal  point  one  place  in  column  D   is  equivalent  to  moving  it  one 
place  in  the  body  of  the  table. 

Circumference  =  ic  X  diam.  =  3.141593  X  diam. 
Conversely, 

Diameter  =  -  X  circumf.  =  0.31831  X  circumf. 


MATHEMATICAL  TABLES 
CIRCUMFERENCES  BY  HUNDREDTHS  (continued) 


29 


D 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

9 

5.0 

15.71 

15.74 

15.77 

15.80 

15.83 

15.87 

15.90 

15.93 

15.% 

15.99 

3 

.1 

16.02 

16.05 

16.08 

16.12 

16.15 

16.18 

16.21 

16.24 

16.27 

16.30 

.2 

16.34 

16.37 

16.40 

16.43 

16.46 

16.49 

16.52 

16.56 

16.59 

16.62 

.3 

16.65 

16.68 

16.71 

16.74 

16.78 

16.81 

16.84 

16.87 

16.90 

16.93 

.4 

16.96 

17.00 

17.03 

17.06 

17.09 

17.12 

17.15 

17.18 

17.22 

17.25 

5.5 

17.28 

17.31 

17.34 

17.37 

17.40 

17.44 

17.47 

17.50 

17.53 

17.56 

j6 

17.59 

17.62 

17.66 

17.69 

17.72 

17.75 

17.78 

17.81 

17.84 

17.88 

.7 

17.91 

17.94 

17.97 

18.00 

18.03 

18.06 

18.10 

18.13 

18.16 

18.19 

.8 

18.22 

18.25 

18.28 

18.32 

18.35 

18.38 

18.41 

18.44 

18.47 

18.50 

.9 

18.54 

18.57 

18.60 

18.63 

18.66 

18.69 

18.72 

18.76 

18.79 

18.82 

6.0 

18.85 

18.88 

18.91 

18.94 

18.98 

19.01 

19.04 

19.07 

19.10 

19.13 

.1 

19.16 

19.20 

19.23 

19.26 

19.29 

19.32 

19.35 

19.38 

19.42 

19.45 

2 

19.48 

19.51 

19.54 

19.57 

19.60 

19.63 

19.67 

19.70 

19.73 

19.76 

3 

19.79 

19.82 

19.85 

19.89 

19.92 

19.95 

19.98 

20.01 

20.04 

20.07 

.4 

20.11 

20.14 

20.17 

20.20 

20.23 

20.26 

20.29 

20.33 

20.36 

20.39 

6.6 

20.42 

20.45 

20.48 

20.51 

20.55 

20.58 

20.61 

20.64 

20.67 

20.70 

.6 

20.73 

20.77 

20.80 

20.83 

20.86 

20.89 

20.92 

20.95 

20.99 

21.02 

.7 

21.05 

21.08 

21.11 

21.14 

21.17 

21.21 

21.24 

21.27 

21.30 

21.33 

.8 

21.36 

21.39 

21.43 

21.46 

21.49 

21.52 

21.55 

21.58 

21.61 

21.65 

.9 

21.68 

21.71 

21.74 

21.77 

21.80 

21.83 

21.87 

21.90 

21.93 

21.96 

7.0 

21.99 

22.02 

22.05 

22.09 

22.12 

22.15 

22.18 

22.21 

22.24 

22.27 

.1 

22.31 

22.34 

22.37 

22.40 

22.43 

22.46 

22.49 

22.53 

22.56 

22.59 

.2 

22.62 

22.65 

22.68 

22.71 

22.75 

22.78 

22.81 

22.84 

22.87 

22.90 

.3 

22.93 

22.97 

23.00 

23.03 

23.06 

23.09 

23.12 

23.15 

23.18 

23.22 

.4 

23.25 

23.28 

23.31 

23.34 

23.37 

23.40 

23.44 

23.47 

23.50 

23.53 

7.5 

23.56 

23.59 

23.62 

23.66 

23.69 

23.72 

23.75 

23.78 

23.81 

23.84 

.6 

23.88 

23.91 

23.94 

23.97 

24.00 

24.03 

24.06 

24.10 

24.13 

24.16 

.7 

24.19 

24.22 

24.25 

24.28 

24.32  • 

24.35 

24.38 

24.41 

24.44 

24.47 

.8 

24.50 

24.54 

24.57 

24.60 

24.63 

24.66 

24.69 

24.72 

24.76 

24.79 

.9 

24.82 

24.85 

24.88 

24.91 

24.94 

24.98 

25.01 

25.04 

25.07 

25.10 

8.0 

25.13 

25.16 

25.20 

25.23 

25.26 

25.29 

25.32 

25.35 

25.38 

25.42 

.1 

25.45 

25.48 

25.51 

25.54 

25.57 

25.60 

25.64 

25.67 

25.70 

25.73 

2 

25.76 

25.79 

25.82 

25.86 

25.89 

25.92 

25.95 

25.98 

26.01 

26.04 

.3 

26.08 

26.11 

26.14 

26.17 

26.20 

26.23 

26.26 

26.30 

26.33 

26.36 

.4 

26.39 

26.42 

26.45 

26.48 

26.52 

26.55 

26.58 

26.61 

26.64 

26.67 

8.5 

26.70 

26.73 

26.77 

26.80 

26.83 

26.86 

26.89 

26.92 

26.95 

26.99 

.6 

27.02 

27.05 

27.08 

27.11 

27.14 

27.17 

27.21 

27.24 

27.27 

27.30 

7 

27.33 

27.36 

27.39 

27.43 

27.46 

27.49 

27.52 

27.55 

27.58 

27.61 

.8 

27.65 

27.68 

27.71 

27.74 

27.77 

27.80 

27.83 

27.87 

27.90 

27.93 

.9 

27.96 

27.99 

28.02 

28.05 

28.09 

28.12 

28.15 

28.18 

28.21 

28.24 

9.0 

28.27 

28.31 

28.34 

28.37 

28.40 

28.43 

28.46 

28.49 

28.53 

28.56 

.1 

28.59 

28.62 

28.65 

28.68 

28.71 

28.75 

28.78 

28.81 

28.84 

28.87 

.2 

28.90 

28.93 

28.97 

29.00 

29.03 

29.06 

29.09 

29.12 

29.15 

29.19 

3 

29.22 

29.25 

29.28 

29.31 

29.34 

29.37 

29.41 

29.44 

29.47 

29.50 

.4 

29.53 

29.56 

29.59 

29.63 

29.66 

29.69 

29.72 

29.75 

29.78 

29.81 

9.5 

29.85 

29.88 

29.91 

29.94 

29.97 

30.00 

30.03 

30.07 

30.10 

30.13 

.6 

30.16 

30.19 

30.22 

30.25 

30.28 

30.32 

30.35 

30.38 

30.41 

30.44 

.7 

30.47 

30.50 

30.54 

30.57 

30.60 

30.63 

30.66 

30.69 

30.72 

30.76 

.8 

30.79 

30.82 

30.85 

30.88 

30.91 

30.94 

30.98 

31.01 

31.04 

31.07 

.9 

31.10 

31.13 

31.16 

31.20 

31.23 

31.26 

31.29 

31.32 

31.35 

31.38 

10.0 

31.42 

Moving  the  decimal  point  ONE  place  in  D  requires  moving  it  ONE  plaoe  in  body  of 
table  (see  p.  28). 


30 


MATHEMATICAL  TABLES 


AREAS  OF  CIRCLES  BY  HUNDREDTHS 

(For  areas  by  eighths,  see  p.  32) 


D 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

?« 

<3 

1.0 

0.785 

0.801 

0.817 

0.833 

0.849 

0.866 

0.882 

0.899 

0.916 

0.933 

16 

.1 

0.950 

0.968 

0.985 

1.003 

1.021 

1.039 

1.057 

1.075 

1.094 

1.112 

18 

.2 

1.131 

1.150 

1.169 

1.188 

1.208 

1.227 

1.247 

1.267 

1.287 

1.307 

20 

.3 

1.327 

1.348 

1.368 

1.389 

1.410 

1.431 

1.453 

1.474 

1.496 

1.517 

21 

A 

1.539 

1.561 

1.584 

1.606 

1.629 

1.651 

1.674 

1.697 

1.720 

1.744 

23 

1.5 

1.767 

1.791 

1.815 

1.839 

1.863 

1.887 

1.911 

1.936 

1.961 

1.986 

24 

.6 

2.011 

2.036 

2.061 

2.087 

2.112 

2.138 

2.164 

2.190 

2.217 

2.243 

26 

.7 

2.270 

2.297 

2.324 

2.351 

2.378 

2.405 

2.433 

2.461 

2.488 

2.516 

27 

.8 

2.545 

2.573 

2.602 

2.630 

2.659 

2.688 

2.717 

2.746 

2.776 

2.806 

29 

.9 

2.835 

2.865 

2.895 

2.926 

2.956 

2.986 

3.017 

3.048 

3.079 

3.110 

31 

2.0 

3.142 

3.173 

3.205 

3.237 

3.269 

3.301 

3.333 

3.365 

3.398 

3.431 

32 

.1 

3.464 

3.497 

3.530 

3.563 

3.597 

3.631 

3.664 

3.698 

3.733 

3.767 

34 

.2 

3.801 

3.836 

3.871 

3.906 

3.941 

3.976 

4.011 

4.047 

4.083 

4.119 

35 

.3 

4.155 

4.191 

4.227 

4.264 

4.301 

4.337 

4.374 

4.412 

4.449 

4.486 

37 

A 

4.524 

4.562 

4.600 

4.638 

4.676 

4.714 

4.753 

4.792 

4.831 

4.870 

38 

2.5 

4.909 

4.948 

4.988 

5.027 

5.067 

5.107 

5.147 

5.187 

5.228 

5.269 

40 

.6 

5.309 

5.350 

5.391 

5.433 

5.474 

5.515 

5.557 

5.599 

5.641 

5.683 

42 

.7 

5.726 

5.768 

5.811 

5.853 

5.896 

5.940 

5.983 

6.026 

6.070 

6.114 

43 

.8 

6.158 

6.202 

6.246 

6.290 

6.335 

6.379 

6.424 

6.469 

6.514 

6.560 

45 

.9 

6.605 

6.651 

6.697 

6.743 

6.789 

6.835 

6.881 

6.928 

6.975 

7.022 

46 

3.0 

7.069 

7.116 

7.163 

7.211 

7.258 

7.306 

7.354 

7.402 

7.451 

7.499 

48 

.1 

7.548 

7.596 

7.645 

7.694 

7.744 

7.793 

7.843 

7.892 

7.942 

7.992 

49 

.2 

8.042 

8.093 

8.143 

8.194 

8.245 

8.296 

8.347 

8.398 

8.450 

8.501 

51 

.3 

8.553 

8.605 

8.657 

8.709 

8.762 

8.814 

8.867 

8.920 

8.973 

9.026 

53 

A 

9.079 

9.133 

9.186 

9.240 

9.294 

9.348 

9.402 

9.457 

9.511 

9.566 

54 

3.5 

9.621 

9.676 

9.731 

9.787 

9.842 

9.898 

9.954 

10.010 

56 

.5 

• 

10.01 

10.07 

10.12 

6 

.6 

10.18 

10.24 

10.29 

10.35 

10.41 

10.46 

10.52 

10.58 

10.64 

10.69 

6 

.7 

10.75 

10.81 

10.87 

10.93 

10.99 

11.04 

11.10 

11.16 

11.22 

11.28 

.8 

11.34 

11.40 

11.46 

11.52 

11.58 

11.64 

11.70 

11.76 

11.82 

11.88 

.9 

11.95 

12.01 

12.07 

12.13 

12.19 

12.25 

12.32 

12.38 

12.44 

12.50 

4.0 

12.57 

12.63 

12.69 

12.76 

12.82 

12.88 

12.95 

13.01 

13.07 

13.14 

7 

.1 

13.20 

13.27 

13.33 

13.40 

13.46 

13.53 

13.59 

13.66 

13.72 

13.79 

.2 

13.85 

13.92 

13.99 

14.05 

14.12 

14.19 

14.25 

14.32 

14.39 

14.45 

.3 

14.52 

14.59 

14.66 

14.73 

14.79 

14.86 

14.93 

15.00 

15.07 

15.14 

A 

15.21 

15.27 

15.34 

15.41 

15.48 

15.55 

15.62 

15.69 

15.76 

15.83 

4.5 

15.90 

15.98 

16.05 

16.12 

16.19 

16.26 

16.33 

16.40 

16.47 

16.55 

.6 

16.62 

16.69 

16.76 

16.84 

16.91 

16.98 

17.06 

17.13 

17.20 

17.28 

.7 

17.35 

17.42 

17.50 

17.57 

17.65 

17.72 

17.80 

17.87 

17.95 

18.02 

.8 

18.10 

18.17 

18.25 

18.32 

18.40 

18.47 

18.55 

18.63 

18.70 

18.78 

8 

.9 

18.86 

18.93 

19.01 

19.09 

19.17 

19.24 

19.32 

19.40 

19.48 

19.56 

Explanation  of  Table  of  Areas  of  Circles  (pp.  30-31) 

Moving  the  decimal  point  one  place  in  column  D  is  equivalent  to  moving  it  two 
places  in  the  body  of  the  table.     (D  =  diameter.) 

Area  of  circle  =  ^  X  (diam.2)  =  0.785398  X  (diam.2) 
Conversely, 


Diam.  = 


=  1.128379  X 


MATHEMATICAL  TABLES 
AREAS  OP  CIRCLES  BY  HUNDREDTHS  (continued) 


31 


D 

0 

1 

2 

a 

4 

5 

6 

7 

8 

9 

$*i 
£-3 

5.0 

19.63 

19.71 

19.79 

19.87 

19.95 

20.03 

20.11 

20.19 

20.27 

20.35 

8 

.1 

20.43 

20.51 

20.59 

20.67 

20.75 

20.83 

20.91 

20.99 

21.07 

21.16 

.2 

21.24 

21.32 

21.40 

21.48 

21.57 

21.65 

21.73 

21.81 

21.90 

21.98 

.3 

22.06 

22.15 

22.23 

22.31 

22.40 

22.48 

22.56 

22.65 

22.73 

22.82 

.4 

22.90 

22.99 

23.07 

23.16 

23.24 

23.33 

23.41 

23.50 

23.59 

23.67 

9 

5.5 

23.76 

23.84 

23.93 

24.02 

24.11 

24.19 

24.28 

24.37 

24.45 

24.54 

.6 

24.63 

24.72 

24.81 

24.89 

24.98 

25.07 

25.16 

25.25 

25.34 

25.43 

.7 

25.52 

25.61 

25.70 

25.79 

25.88 

25.97 

26.06 

26.15 

26.24 

26.33 

.8 

26.42 

26.51 

26.60 

26.69 

26.79 

26.88 

26.97 

27.06 

27.15 

27.25 

.9 

27.34 

27.43 

27.53 

27.62 

27.71 

27.81 

27.90 

27.99 

28.09 

28.18 

6.0 

28.27 

28.37 

28.46 

28.56 

28.65 

28.75 

28.84 

28.94 

29.03 

29.13 

10 

.1 

29.22 

29.32 

29.42 

29.51 

29.61 

29.71 

29.80 

29.90 

30.00 

30.09 

.2 

30.19 

30.29 

30.39 

30.48 

30.58 

30.68 

30.78 

30.88 

30.97 

31.07 

.3 

31.17 

31.27 

31.37 

31.47 

31.57 

31.67 

31.77 

31.87 

31.97 

32.07 

.4 

32.17 

32.27 

32.37 

32.47 

32.57 

32.67 

32.78 

32.88 

32.98 

33.08 

6.5 

33.18 

33.29 

33.39 

33.49 

33.59 

33.70 

33.80 

33.90 

34.00 

34.11 

.6 

34.21 

34.32 

34.42 

34.52 

34.63 

34.73 

34.84 

34.94 

35.05 

35.15 

.7 

35.26 

35.36 

35.47 

35.57 

35.68 

35.78 

35.89 

36.00 

36.10 

36.21 

11 

.8 

36.32 

36.42 

36.53 

36.64 

36.75 

36.85 

36.96 

37.07 

37.18 

37.28 

.9 

37.39 

37.50 

37.61 

37.72 

37.83 

37.94 

38.05 

38.16 

38.26 

38.37 

7.0 

38.48 

38.59 

38.70 

38.82 

38.93 

39.04 

39.15 

39.26 

39.37 

39.48 

.1 

39.59 

39.70 

39.82 

39.93 

40.04 

40.15 

40.26 

40.38 

40.49 

40.60 

.2 

40.72 

40.83 

40.94 

41.06 

41.17 

41.28 

41.40 

41.51 

41.62 

41.74 

.3 

41.85 

41.97 

42.08 

42.20 

42.31 

42.43 

42.54 

42.66 

42.78 

42.89 

12 

.4 

43.01 

43.12 

43.24 

43.36 

43.47 

43.59 

43.71 

43.83 

43.94 

44.06 

7.5 

44.18 

44.30 

44.41 

44.53 

44.65 

44.77 

44.89 

45.01 

45.13 

45.25 

.6 

45.36 

45.48 

45.60 

45.72 

45.84 

45.96 

46.08 

46.20 

46.32 

46.45 

.7 

46.57 

46.69 

46.81 

46.93 

47.05 

47.17 

47.29 

47.42 

47.54 

47.66 

.8 

47.78 

47.91 

48.03 

48.15 

48.27 

48.40 

48.52 

48.65 

48.77 

48.89 

.9 

49.02 

49.14 

49.27 

49.39 

49.51 

49.64 

49.76 

49.89 

50.01 

50.14 

8.0 

50.27 

50.39 

50.52 

50.64 

50.77 

50.90 

51.02 

51.15 

51.28 

51.40 

13 

.1 

51.53 

51.66 

51.78 

51.91 

52.04 

52.17 

52.30 

52.42 

52.55 

52.68 

.2 

52.81 

52.94 

53.07 

53.20 

53.33 

53.46 

53.59 

53.72 

53.85 

53.98 

.3 

54.11 

54.24 

54.37 

54.50 

54.63 

54.76 

54.89 

55.02 

55.15 

55.29 

.4 

55.42 

55.55 

55.68 

55.81 

55.95 

56.08 

56.21 

56.35 

56.48 

56.61 

8.5 

56.75 

56.88 

57.01 

57.15 

57.28 

57.41 

57.55 

57.68 

57.82 

57.95 

.6 

58.09 

58.22 

58.36 

58.49 

58.63 

58.77 

58.90 

59.04 

59.17 

59.31 

14 

.7 

59.45 

59.58 

59.72 

59.86 

59.99 

60.13 

60.27 

60.41 

60.55 

60.68 

.8 

60.82 

60.96 

61.10 

61.24 

61.38 

61.51 

61.65 

61.79 

61.93 

62.07 

.9 

62.21 

62.35 

62.49 

62.63 

62.77 

62.91 

63.05 

63.19 

63.33 

63.48 

9.0 

63.62 

63.76 

63.90 

64.04 

64.18 

64.33 

64.47 

64.61 

64.75 

64.90 

.1 

65.04 

65.18 

65.33 

65.47 

65.61 

65.76 

65.90 

66.04 

66.19 

66.33 

15 

.2 

66.48 

66.62 

66.77 

66.91 

67.06 

67.20 

67.35 

67.49 

67.64 

67.78 

.3 

67.93 

68.08 

68.22 

68.37 

68.51 

68.66 

68.81 

68.96 

69.10 

69.25 

.4 

69.40 

69.55 

69.69 

69.84 

69.99 

70.14 

70.29 

70.44 

70.58 

70.73 

9.5 

70.88 

71.03 

71.18 

71.33 

71.48 

71.63 

71.78 

71.93 

72.08 

72.23 

.6 

72.38 

72.53 

72.68 

72.84 

72.99 

73.14 

73.29 

73.44 

73.59 

73.75 

.7 

73.90 

74.05 

74.20 

74.36 

74.51 

74.66 

74.82 

74.97 

75.12 

75.28 

.8 

75.43 

75.58 

75.74 

75.89 

76.05 

76.20 

76.36 

76.51 

76.67 

76.82 

.9 

76.98 

77.13 

77.29 

77.44 

77.60 

77.76 

77.91 

78.07 

78.23 

78.38 

16 

Moving  the  decimal  point  ONE  place  in  D  requires  moving  it  TWO  places  in  body 
of  table  (see  p.  30). 


32 


MATHEMATICAL  TABLES 


CIRCUMFERENCES  AND  AREAS  OF  CIRCLES  BY  EIGHTHS,  ETC. 

(For  tenths,  see  p.  28) 


1 

| 

1 

S 
• 

a 

Circum. 

1 

I 

Circum. 

a 
£ 

•< 

§ 

Q 

Circum. 

ot 

2 
<5 

H 

2.749 

.6013 

4 

12.57 

12.57 

9 

28.27 

63.62 

W4 

.04909 

.00019 

6%4 

2.798 

.6230 

He 

12.76 

12.96 

H 

28.67 

65.40 

fa 

.09817 

.00077 

*fa 

2.847 

.6450 

H 

12.96 

13.36 

H 

29.06 

67.20 

fa 

.1473 

.00173 

*%* 

2.896 

.6675 

Me 

13.16 

13.77 

N 

29.45 

69.03 

Ho 

.1963 

.00307 

'Me 

2.945 

.6903 

H 

13.35 

14.19 

H 

29.85 

70.88 

%« 

.2454 

.00479 

6^4 

2.994 

.7135 

Me 

13.55 

14.61 

K^ 

30.24 

7276 

fa 

.2945 

.00690 

•fa 

3.043 

.7371 

13.74 

15.03 

y\ 

30.63 

74.66 

ft*. 

.3436 

.00940 

*%l 

3.093 

.7610 

Mo 

13.94 

15.47 

1& 

31.02 

76.59 

H 

.3927 

.01227 

1 

3.142 

.7854 

H 

14.14 

15.90 

10 

31.42 

78.54 

Hi 

.4418 

.01553 

Me 

3.338 

.8866 

Me 

14.33 

16.35 

H 

31.81 

80.52 

fa 

.4909 

.01917 

H 

3.534 

.9940 

5/i 

14.53 

16.80 

H 

32.20 

82.52 

'^4 

.5400 

.02320 

Me 

3.731 

1.108 

'He 

14.73 

17.26 

H 

32.59 

84.54 

Me 

.5890 

.02761 

U 

3.927 

1.227 

N 

14.92 

17.72 

M 

32.99 

86.59 

'%4 

.6381 

.03241 

Me 

4.123 

1.353 

'Me 

15.12 

18.19 

M 

33.38 

88.66 

fa 

.6872 

.03758 

M 

4.320 

1.485 

H 

15.32 

18.67 

94 

33.77 

90.76 

'5*4 

.7363 

.04314 

Me 

4.516 

1.623 

'Me 

15.51 

19.15 

H 

34.16 

92.89 

u 

.7854 

.04909 

H 

4.712 

1.767 

5 

15.71 

19.63 

11 

34.56 

95.03 

»%4 

.8345 

.05542 

Me 

4.909 

1.917 

He 

15.90 

20.13 

M 

34.95 

97.21 

fa 

.8836 

.06213 

5/6 

5.105 

2.074 

H 

16.10 

20.63 

H 

35.34 

99.40 

l%i 

.9327 

.06922 

'He 

5.301 

2.237 

Me 

16.30 

21.14 

N 

35.74 

101.6 

Me 

.9817 

.07670 

N 

5.498 

2.405 

M 

16.49 

21.65 

H 

36.13 

103.9 

2^4 

1.031 

.08456 

'Me 

5.694 

2.580 

Me 

16.69 

22.17 

% 

36.52 

106.1 

Ifa 

1.080 

.09281 

H 

5.890 

2.761 

N 

16.89 

22.69 

H 

36.91 

108.4 

2*<J4 

1.129 

.1014 

I?i8 

6.087 

2.948 

Me 

17.08 

23.22 

N 

37.31 

110.8 

H 

1.178 

.1104 

a 

6.283 

3.142 

W 

17.28 

23.76 

12 

37.70 

113.1 

*M* 

1.227 

.1198 

He 

6.480 

3.341 

Me 

17.48 

24.30 

H 

38.09 

115.5 

lfa 

1.276 

.1296 

H 

6.676 

3.547 

M 

17.67 

24.85 

H 

38.48 

117.9 

2  ^4 

1.325 

.1398 

Me 

6.872 

3.758 

'He 

17.87 

25.41 

% 

38.88 

120.3 

Me 

1.374 

.1503 

M 

7.069 

3.976 

H 

18.06 

25.97 

M 

39.27 

122.7 

2%4 

1.424 

.1613 

Me 

7.265 

4.200 

'Me 

18.26 

26.53 

U 

39.66 

125.2 

»9$2 

1.473 

.1726 

7.461 

4.430 

N 

18.46 

27.11 

M- 

40.06 

127.7 

»H* 

1.522 

.1843 

Me 

7.658 

4.666 

'Me 

18.65 

27.69 

ji 

40.45 

130.2 

M 

1.571 

.1963 

W 

7.854 

4.909 

6 

18.85 

28.27 

13 

40.84 

132.7 

8%4 

1.620 

.2088 

Me 

8.050 

5.157 

M 

19.24 

29.46 

w 

41.23 

135.3 

lfa 

1.669 

.2217 

H 

8.247 

5.412 

M 

19.63 

30.68 

H 

41.63 

137.9 

•fa 

1.718 

.2349 

»H« 

8.443 

5.673 

N 

20.03 

31.92 

N 

42.02 

140.5 

Me 

1.767 

.2485 

94 

8.639 

5.940 

H- 

20.42 

33.18 

H 

42.41 

143.1 

3%4 

1.816 

.2625 

me 

8.836 

6.213 

M 

20.81 

34.47 

W 

42.80 

145.8 

lfa 

1.865 

.2769 

H 

9.032 

6.492 

M 

21.21 

35.78 

H 

43.20 

148.5 

»%4 

1.914 

.2916 

'Me 

9.228 

6.777 

N 

21.60 

37.12 

7/i 

43.59 

151.2 

M 

1.963 

.3068 

3 

9.425 

7.069 

7 

21.99 

38.48 

14 

43.98 

153.9 

*H4 

2.013 

.3223 

He 

9.621 

7.366 

K 

22.38 

39.87 

M 

44.37 

156.7 

9  fa 

2.062 

.3382 

H 

9.817 

7.670 

M 

22.78 

41.28 

M 

44.77 

159.5 

*%4 

2.111 

.3545 

Me 

10.01 

7.980 

n 

23.17 

42.72 

N 

45.16 

162.3 

'Me 

2.160 

.3712 

H 

10.21 

8.296 

H 

23.56 

44.18 

H 

45.55 

165.1 

4$B4 

2.209 

.3883 

Me 

10.41 

8.618 

% 

23.95 

45.66 

H 

45.95 

168.0 

2  fa 

2.258 

.4057 

M 

10.60 

8.946 

H 

24.35 

47.17 

N 

46.34 

170.9 

*K* 

2.307 

.4236 

Me 

10.80 

9.281 

X 

24.74 

48.71 

H 

46.73 

173.8 

H 

2.356 

.4418 

H 

11.00 

9.621 

8 

25.13 

50.27 

15 

47.12 

176.7 

*%4 

2.405 

.4604 

Me 

11.19 

9.968 

H 

25.53 

51.85 

H 

47.52 

179.7 

'fa 

2.454 

.4794 

#* 

11.39 

10.32 

M 

25.92 

53.46 

H 

47.91 

182.7 

«H4 

2.503 

.4987 

'He 

11.58 

10.68 

H 

26.31 

55.09 

H 

48.30 

185.7 

'Me 

2.553 

.5185 

N 

11.78 

11.04 

H 

26.70 

56.75 

M 

48.69 

188.7 

3%4 

2.602 

.5386 

'Me 

11.98 

11.42 

N 

27.10 

58.43 

H 

49.09 

191.7 

*fa 

2.651 

.5591 

H 

12.17 

11.79 

N 

27.49 

60.13 

N 

49.48 

194.8 

•%4 

2.700 

.5800 

i«s 

12.37 

12.18 

N 

27.88 

61.86 

K 

49.87 

197.9 

MATHEMATICAL  TABLES 


33 


CIRCUMFERENCES  AND  AREAS  BY  EIGHTHS— (continued) 


s 

.5 
P 

Circum. 

1 

P 

j 

a 

| 

Circum. 

1 

a 
P 

Circum. 

1 

16 

50.27 

201.1 

19  H 

61.26 

298.6 

23 

7226 

415.5 

29 

91.11 

660.5 

50.66 

204.2 

H 

61.65 

302.5 

H 

72.65 

420.0 

H 

91.89 

672.0 

H 

51.05 

207.4 

N 

62.05 

306.4 

73.04 

424.6 

N 

92.68 

683.5 

N 

51.44 

210.6 

R 

62.44 

310.2 

H 

73.43 

429.1 

H 

93.46 

695.1 

M 

51.84 

213.8 

20 

6283 

314.2 

H 

73.83 

433.7 

30 

94.25 

706.9 

S£ 

52.23 

217.1 

H 

6322 

318.1 

7422 

438.4 

H 

95.03 

718.7 

3/ 

52.62 

220.4 

H 

63.62 

322.1 

?i 

74.61 

443.0 

H 

95.82 

730.6 

£i 

53.01 

223.7 

N 

64.01 

326.1 

8 

75.01 

447.7 

N 

96.60 

742.6 

17 

53.41 

227.0 

M 

64.40 

330.1 

24 

75.40 

452.4 

31 

97.39 

754.8 

H 

53.80 

230.3 

N 

64.80 

334.1 

M 

76.18 

461.9 

H 

98.17 

767.0 

H 

54.19 

233.7 

65.19 

3382 

M 

76.97 

471.4 

M 

98.96 

779.3 

N 

54.59 

237.1 

% 

65.58 

3422 

N 

77.75 

481.1 

§4 

99.75 

791.7 

M 

54.98 

240.5 

21 

65.97 

346.4 

25 

78.54 

490.9 

32 

100.5 

804.2 

&4 

55.37 

244.0 

H 

66.37 

350.5 

H 

79.33 

500.7 

H 

101.3 

816.9 

a/ 

55.76 

247.4 

M 

66.76 

354.7 

80.11 

510.7 

h 

102.1 

829.6 

H 

56.16 

250.9 

34 

67.15 

358.8 

N 

80.90 

520.8 

N 

102.9 

842.4 

18 

56.55 

254.5 

M 

67.54 

363.1 

26 

81.68 

530.9 

33 

103.7 

855.3 

M 

56.94 

258.0 

W 

67.94 

367.3 

M 

82.47 

541.2 

M 

104.5 

8683 

H 

57.33 

261.6 

68.33 

371.5 

8325 

551.5 

1052 

881.4 

N 

57.73 

265.2 

j| 

68.72 

375.8 

H 

84.04 

562.0 

% 

106.0 

894.6 

M 

58.12 

268.8 

22 

69.12 

380.1 

27 

84.82 

572.6 

34 

106.8 

907.9 

8 

58.51 
58.90 

272.4 
276.1 

g 

69.51 
69.90 

384.5 
388.8 

*5 

85.61 
86.39 

583.2 
594.0 

i 

107.6 
108.4 

921.3 
934.8 

J6 

59.30 

279.8 

N 

7029 

393.2 

N 

87.18 

604.8 

109.2 

948.4 

19 

59.69 

283.5 

fc 

70.69 

397.6 

28 

87.96 

615.8 

35 

110.0 

962.1 

H 

60.08 

287.3 

71.08 

402.0 

H 

88.75 

626.8 

5-4 

110.7 

975.9 

U 

60.48 

291.0 

?4 

71.47 

406.5 

B 

89.54 

637.9 

M 

111.5 

989.8 

N 

60.87 

294.8 

N 

71.86 

411,0 

90.32 

6492 

N 

112.3 

1003.8 

AREAS  OF  CIRCLES.     Diameters  in  Feet  and  Inches,  Areas  in  Square  Feet 


Feet 

Inches 

0           1           23456           789          10 

11 

0 

1 
2 
3 
4 

5 

6 
7 
8 
? 
10 
11 
12 
13 
14 

.0000    .0055    .0218    .0491     .0873    .1364    .1963    .2673    .3491     .4418    .5454 
.7854    .9218    1.069    1.227    1.396    1.576    1.767    1.969    2.182    2.405    2.640 
3.142    3.409    3.687    3.976    4.276    4.587    4.909    5.241    5.585    5.940    6.305 
7.069    7.467    7.876    8.2%    8.727    9.168    9.621     10.08    10.56    11.04    11.54 
12.57    13.10    13.64    14.19    14.75    15.32    15.90    16.50    17.10    1772    18.35 

19.63    20.29    20.97    21.65    22.34    23.04    23.76    24,48    2522    25.97    26.73 
28.27    29.07    29.87    30.68    31.50    32.34    33.18    34.04    34.91    35.78    36.67 
38.48    39.41    40.34    41.28    42.24    43.20    44.18    45.17    46.16    47.17    48.19 
50.27    51.32    52.38    53.46    54.54    55.64    56.75    57.86    58.99    60.13    61.28 
63.62    64.80    66.00    6720    68.42    69.64    70.88    72.13    73.39    74.66    75.94 

78.54    79.85    81.18    82.52    83.86    85.22    86.59    87.97    89.36    90.76    92.18 
95.03    96.48    97.93    99.40    100.9    102.4    103.9    105.4    106.9    108.4    110.0 
113.1     114.7    116.3    117.9    119.5    121.1     122.7    124.4    126.0    127.7    129.4 
132.7    134.4    136.2    137.9    139.6    141.4    143.1     144.9    146.7    148.5    150.3 
153.9    155.8    157.6    1593    161.4    1632    165.1     167.0    168.9    170.9    172.8 

.6600 
2.885 
6.681 
12.05 
18.99 

27.49 
37.57 
4922 
62.44 
7724 

93.60 
111.5 
131.0 
152.1 
174.8 

If  given  diameter  is  not  found  in  this  table,  reduce  diameter  to  feet  and  decimals  of  a 
foot  by  aid  of  the  following  auxiliary  table,  and  then  find  area  from  pp.  30-31. 

From  Inches  and  Fractions  of  an  Inch  to  Decimals  of  a  Foot 

Inches             123           456           78           9         10 
Feet             .0833  .1667  .2500  .3333  .4167  .5000  .5833   .6667  .7500  .8333 

11 
.9167 

Inches            H         H         H          M         5/i         9*         7/* 
Feet             .0104  .0208  .0313  .0417  .0521  .0625  .0729 
Example.    5  ft.  7%  in.  =  5.0  +  0.5833  +  0.0313  =  5.6146  ft. 

34 


MATHEMATICAL  TABLES 


SEGMENTS  OP  CIRCLES,  GIVEN  h/c 

Given:  h  =  height;  c  =  chord.     (For  explanation  of  this  table,  see  p.  38) 


h 

Diam.   ja 

Arc    « 

Area    £: 

Central   itt 

A      te 

c 

c     P 

c     P 

h  X  c   p 

angle,  »   Q 

Diam.   P 

.00 

1.000     n 

.6667     n 

0.00°    4,0 

.0000 

25.010   174gn 

1.000     0 

.6667 

4.58 

.0004    ,1 

2 

19  s?n   i*^"u 

I/.J/U     *A1C7 

1.001 

.6669     ± 

9  .6 

.00.6   ;J 

3 

fl  3A3    4  1  J/ 
o.?05    *7f)7^ 

1.002     i 

.667. 

13J3    I5.7, 

.0036    ?J 

4 

6.290   .f073 

1.004     £ 

.6675     I 

18.30    Jg 

.0064    g 

.05 

5.050    ,,0,3 

1.007 

.6680     , 

22.84°    4,3 

.0099 

6 

4.227    »|g 

1.010     \ 

.6686     2 

27.37 

.0142 

7 

3.641    *X7A 

1.013 

.6693     J 

31.88    1!I 

.0192    22 

8 

3.205    ,£7 

1.017 

.670. 

36.36 

.0250    22 

9 

2.868    ^g 

1.021     J 

.6710    |0 

40.82    JJ5 

.0314    64 

.10 

2.600    »,,7 

1.026 

.6720 

45.24°    430 

.0385    77 

1 

2.383    »fl£ 

1.032 

.6731    11 

49.63    1|? 

.0462    o< 

2 

2.203    Jt2 

1.038 

.6743     ? 

53.98    I3* 

.0545    2^ 

3 

2.053    '55 

1.044     2 

.6756     J 

58.30    1^ 

.0633 

4 

1.926    ^ 

1.051     I 

.6770    {J 

62.57    J27 

.0727    99 

.15 

1.817    ,04 

1.059 

.6785 

66.80°    4,0 

.0826    ,03 

6 

.723    ,25 

1.067 

.6801     2 

70.98    J  ? 

.0929    XT' 

7 

1.641     JE 

1.075 

.6818     £ 

75.11    7A^ 

.1036    VJ 

8 

1.569    2? 

1.084    ,2 

.6836 

79.20    1X4 

.1147     J 

9 

1.506     g 

1.094    ^ 

.6855    JJ 

83.23    $5 

•'262    jg 

.20 

1.450     ,n 

1.103    n 

.6875    71 

87.21°    30, 

.1379    nn 

1 

1.400 

1.114     J 

.6896    ?,7 

91.13    S= 

.1499    20 

2 

1.356 

1.124 

.6918    5? 

95.00    W. 

.1622    Jg 

3 

1.317     H 

1.136     2 

.6941    g 

98.81    2J 

.1746    g 

4 

1.282     % 

1.147    » 

.6965    24 

102.56    375 

.1873    Jg 

.25 

6 

1.250     -« 
.222     28 

1.159    12 
1.171    \\ 

.6989    9c 
.7014    ;7 

106.26°    3,4 
109.90    $52 

•2000    ,28 

7 

.196     ?6 

1.184 

.7041    5i 

113.48    358 

^2258    !2J 

8 

.173     23 

1.197 

.7068    11 

117.00    ?52 

.2387    aj 

9 

.152     J. 

1.2.1    JJ 

.7096    g 

120.45    345 

JI517    J30 

.30 

1 
2 
3 

.133     ,7 
.116      i 
.101 
.088 

1.225    ,4 
1.239     7 
1.254    \l 
1.269 

.7125    9q 
.7154    29 
.7185    ^ 
.7216    3,1 

123.86°    334 
127.20    334 
130.48    325 
133.70    322 

.2647    130 
.2777    ,o 
.2906    HJ 
.3034    52 

4 

1.075     jf 

1.284    ]£ 

.7248    55 

136.86    316 

3162    Jg 

.35 

1.064       ,n 

1.300 

.7280    34 

139.97°    3rt, 

.3289    ,-- 

6 

1.054     '° 

1.316 

.7314    5x 

143.02    305 

.3414    Jg 

7 

1.046 

1.332     2 

.7348    g 

146.01    ££ 

.3538    \£ 

8 

1.038     2 

1.349     i 

.7383    ?5 

148.94    ??2 

.3661    g 

9 

1.031      J 

1.366    \'7 

.7419    g 

151.82    288 

3783    J22 

.40 

1.025 
1.020     i 

1.383    ,o 
1.401 

.7455    37 
.7492    on 

154.64°    -77 
157.41    El 

4022    "9 

.W/l     jj> 

2 

1.015 

1.419 

.7530    IS 

160.12    %\ 

.4137    JJfj 

3 

1.011 

1.437 

.7568    3*5 

162.78    5?? 

4 

1.008     \ 

1.455    J« 

.7607    ^Q 

165.39    261 

.4364    jj2 

.45 

6 

8 
9 

1.006 
1.003 
1.002 
1.001 
1.000     J 

1.474    ,Q 
1.493 
1.512 
1.531    ij 
1.55.    20 

.7647    40 
.7687    TV 
.7728    J 
.7769    11 
.781.    « 

167.95°    ,,, 
170.46    gi 
172.91    fjf 
175.32    241 
177.69    g7 

.4475    )no 
.4584    X^ 
.4691    2? 
.4796    25 
.4899    JJJ 

.50 

1.000 

1.571 

.7854 

180.00° 

.5000 

•  Interpolation  may  be  inaccurate  at  these  points. 


MATHEMATICAL  TABLES 


.SEGMENTS  OP  CIRCLES,  GIVEN  h/D 

Given:  h  «=•  height;  D  =  diameter  of  circle.    (For  explanation  of  this  table,  see  p.  38) 


Arc 


Area 


Central  i 
angle,  v 


Chord 


Arc 


Circumf.  Q 


Area 
Ckde 


2003  2003 

3482  »t1c 

.4027  ,Jg 

.4510  *4™ 

.4949  »;£: 

.5355  Jftn 
5735 

.'6094  'U9 

.6435  *„, 

.676.  Jf6 

.7075  JA; 

.7377  JJ2 

.7670  ,293 

.7954  276 

.8230  2;x 

.8500  ^V 

.8763  ;K 

.9021  g* 

0.9273  24R 

0.9521  215 

0.9764  %£ 

1.0004  i?? 

1.0239  Si 


1.0472 
1.070. 
..0928 
...152 
1.1374 

1.1593 
1.1810 
1.2025 
1.2239 
1.2451 

1.2661 
1.2870 
1.3078 
1.3284 

1.3694 
1.3898 
1.4101 
1.4303 
1.4505 

1.4706 
1.4907 
1.5108 
1.5308 
1.5508 

1.5708 


22g 


222 
219 

217 


212 


202 
2J2 


2 


.0000 
.0013 
.0037 
.0069 
.0105 

.0.47 
.0192 
.0242 
.0294 
.0350 

.0409 
.0470 
.0534 
.0600 
.0668 

.0739 
.0811 
.0885 
.0961 
.1039 

.1118 
.1199 
.128. 
.1365 
.1449 

.1535 
.1623 
.1711 
.1800 
.1890 

.1982 
.2074 
.2167 
.2260 
.2355 

.2450 
.2546 
.2642 
.2739 
.2836 

.2934 
.3032 
.3130 
.3229 
.3328 

.3428 
.3527 

3727 
3827 

3927 


., 


, 


, 

IUU 


0.00°  770, 

22.96  A296 

32.52  956 

39.90  738 

46.15  ,|g 

51-68°  .... 

56.72  504 

6137  *£ 

65.72  435 

69.83  ^J] 

73.74°  „_. 

77.48  *|74 

81.07  359 

84.54  *347 

87.89  .335 

91.15°  ,.., 

94.31 

97.40  309 

.00.42  302 

103.37  295 

106.26° 

109.10  284 

111.89  279 
114.63  274 
11734  27. 

266 

.20.00° 

122.63  263 

125.23  260 

127.79  256 

.3033  254 

132.84° 

.35.33  249 

.37.80  247 

140.25  245 
.42.67  2J2 

145.08° 

.47.48  240 

.49.86  238 

152.23  237 

154.58  235 

156.93°  , 

159.26  233 

161.59  233 

163.90  23. 
166.22  232 

168.52°  ... 

.70.82  230 

173.12  230 

175.42  230 

177.71  229 


180.00° 


.0000 
!2800 
3919 
.4359 

!5I03 
.5426 
.5724 

.6000 
.6258 
.6499 
.6726 
.6940 

.7141 

>513 
.7684 


.8000 
.8146 
.8285 
.8417 

.8660 
.8773 
.8879 
.8980 
.9075 

.9165 
.9250 
.9330 
.9404 
.9474 

.9539 
.9600 
.9656 
.9708 
.9755 

.9798 
.9837 
.9871 
.9902 
.9928 

.9950 
.9968 
.9982 
.9992 
.9998 

1.0000 


3.3 
.298 


241 
J27 
.2.4 


.0000 
.0638 
.0903 
.1108 
.1282 

.1436 

!l705 
.1826 
.1940 

.2048 
.2.52 
.2252 
.2348 
.244. 

.2532 
.2620 
.2706 
.2789 
.2871 

.2952 
3031 
3108 
.3184 
3259 

.3333 
.3406 
3478 
.3550 
.3620 

3690 
3759 
3828 
3896 
.3963 

.4030 
.4097 
.4.63 
.4229 
.4294 

.4359 
.4424 
.4489 
.4553 
.4617 

.4681 
.4745 
.4809 
.4873 
.4936 

.5000 


j{j 


.0000  ,7 

.0017  \\ 

.0048  l\ 

.0087  ?? 

.0134  g 

.0187  «, 

.0245  g 

.0308  g 

.0375  S{ 

.0446  £ 

.0520  TO 

.0599  JJ 
.0680 

.0764  2; 

.085.  §J 

.0941  Q, 

.1033  2 

.1127  A? 

.1224  II 

.1323  ,99 

•!$  103 

.1631  !°< 

.1737  2$ 

.1846  JJ5 

.1955  m 

.2066  1 

.2.78  J 

.2292  \\i 

2407  \\l 

.2523  ,,7 

•2640  £ 

.2759  9 

.2878  ,« 

.2998  ]^° 

.3.19  ,22 

324.  22 

3364  S 

3487  g 

.3611  j|J 

3735  ,« 

3860  g 

3986  \% 

.41.2  5? 

.4238  Jg 

.4364  .„ 

.4491  % 

.4618  ?i 

.4745  g 

.4873  J28 

.5000 


Interpolation  may  be  inaccurate  at  these  points. 


36  MATHEMATICAL  TABLES 

VOLUMES  OF  SPHERES  BY  HUNDREDTHS 


D 

0 

1 

2 

3 

4 

5 

& 

1.0 

.5236 

.5395 

.5556 

.5722 

.5890 

.6061 

.6236 

.6414 

.65% 

.6781 

173 

.6969 

.7161 

.7356 

.7555 

.7757 

.7963 

.8173 

.8386 

.8603 

.8823 

208 

'.2 

.9048 

.9276 

.9508 

.9743 

.9983 

1.0227 

236 

.2 

1.023 

1.047 

1.073 

1.098 

1.124 

25 

.3 

1.150 

1.177 

1.204 

1.232 

1.260 

1.288 

1.317 

1.346 

1.376 

1.406 

29 

.4 

1.437 

1.468 

1.499 

1.531 

1.563 

1.596 

1.630 

1.663 

1.697 

1.732 

33 

1.5 

1.767 

1.803 

1.839 

1.875 

1.912 

1.950 

1.988 

2.026 

2.065 

2.105 

38 

.6 

2.145 

2.185 

2.226 

2.268 

2.310 

2.352 

2.395 

2.439 

2.483 

2.527 

43 

.7 

2.572 

2.618 

2.664 

2.711 

2.758 

2.806 

2.855 

2.903 

2.953 

3.003 

48 

8 

3.054 

3.105 

3.157 

3.209 

3.262 

3.315 

3.369 

3.424 

3.479 

3.535 

54 

.9 

3.591 

3.648 

3.706 

3.764 

3.823 

3.882 

3.942 

4.003 

4.064 

4.126 

60 

2.0 

4.189 

4.252 

4.316 

4.380 

4.445 

4.511 

4.577 

4.644 

4.712 

4.780 

66 

.1 

4.849 

4.919 

4.989 

5.0CO 

5.131 

5.204 

5.277 

5.350 

5.425 

5.500 

73 

.2 

5.575 

5.652 

5.729 

5.806 

5.885 

5.964 

6.044 

6.125 

6.206 

6.288 

80 

.3 

6.371 

6.454 

6.538 

6.623 

6.709 

6.795 

6.882 

6.970 

7.059 

7.148 

87 

.4 

7.238 

7.329 

7.421 

7.513 

7.606 

7.700 

7.795 

7.890 

7.986 

8.083 

94 

2.5 

8.181 

8.280 

8.379 

8.479 

8.580 

8.682 

8.785 

8.888 

8.992 

9.097 

102 

.6 

9.203 

9.309 

9.417 

9.525 

9.634 

9.744 

9.855 

9.966 

10.079 

110 

.6 

10.08 

10.19 

11 

.7 

10.31 

10.42 

10.54 

10.65 

10.77 

10.89 

11.01 

11.13 

11.25 

11.37 

12 

.8 

11.49  x 

11.62 

11.74 

11.87 

11.99 

12.12 

12.25 

12.38 

12.51 

12.64 

13 

.9 

12.77 

12.90 

13.04 

13.17 

13.31 

13.44 

13.58 

13.72 

13.86 

14.00 

14 

3.0 

14.14 

14.28 

14.42 

14.57 

14.71 

14.86 

15.00 

15.15 

15.30 

15.45 

15 

.1 

15.60 

15.75 

15.90 

16.06 

16.21 

16.37 

16.52 

16.68 

16.84 

17.00 

16 

.2 

17.16 

17.32 

17.48 

17.64 

17.81 

17.97 

18.14 

18.31 

18.48 

18.65 

17 

.3 

18.82 

18.99 

19.16 

19.33 

19.51 

19.68 

19.86 

20.04 

20.22 

20.40 

18 

.4 

20.58 

20.76 

20.94 

21.13 

21.31 

21.50 

21.69 

21.88 

22.07 

22.26 

19 

3.5 

22.45 

22.64 

22.84 

23.03 

23.23 

23.43 

23.62 

23.82 

24.02 

24.23 

20 

.6 

24.43 

24.63 

24.84 

25.04 

25.25 

25.46 

25.67 

25.88 

26.09 

26.31 

21 

.7 

26.52 

26.74 

26.95 

27.17 

27.39 

27.61 

27.83 

28.06 

28.28 

28.50 

22 

.8 

28.73 

28.% 

29.19 

29.42 

29.65 

29.88 

30.11 

30.35 

30.58 

30.82 

23 

.9 

31.06 

31.30 

31.54 

31.78 

32.02 

32.27 

32.52 

32.76 

33.01 

33.26 

25 

4.0 

33.51 

33.76 

34.02 

34.27 

34.53 

34.78 

35.04 

35.30 

35.56 

35.82 

26 

.1 

36.09 

36.35 

36.62 

36.88 

37.15 

37.42 

37.69 

37.97 

38.24 

38.52 

27 

.2 

38.79 

39.07 

39.35 

39.63 

39.91 

40.19 

40.48 

40.76 

41.05 

41.34 

28 

.3 

41.63 

41.92 

42.21 

42.51 

42.80 

43.10 

43.40 

43.70 

44.00 

44.30 

30 

.4 

44.60 

44.91 

45.21 

45.52 

45.83 

46.14 

46.45 

46.77 

47.08 

47.40 

31 

4.5 

47.71 

48.03 

48.35 

48.67 

49.00 

49.32 

49.65 

49.97 

50.30 

50.63 

33 

.6 

50.97 

51.30 

51.63 

51.97 

52.31 

52.65 

52.99 

53.33 

53.67 

54.02 

34 

.7 

54.36 

54.71 

55.06 

55.41 

55.76 

56.12 

56.47 

56.83 

57.19 

57.54 

35 

.8 

57.91 

58.27 

58.63 

59.00 

59.37 

59.73 

60.10 

60.48 

60.85 

61.22 

37 

.9 

61.60 

61.98 

62.36 

62.74 

63.12 

63.51 

63.89 

64.28 

64.67 

65.06 

38 

Explanation  of  Table  of  Volumes  of  Spheres  (pp.  36-37). 

Moving  the  decimal  point  one  place  in  column  D  is  equivalent  to  moving  it  three 
places  in  the  body  of  the  table.     (D  =  diameter.) 


Volume  of  sphere  =  7  X  (diarn.»)  =  0.523599  X  (diam.») 


Conversely, 


Diam. 


1.240701  X 


MATHEMATICAL  TABLES 


37 


VOLUMES  OF  SPHERES   (continued) 


D 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

11 

5.0 

65.45 

65.84 

66.24 

66.64 

67.03 

67.43 

67.83 

68.24 

68.64 

69.05 

40 

.1 

69.46 

69.87 

70.28 

70.69 

71.10 

71.52 

71.94 

72.36 

72.78 

73.20 

42 

.2 

73.62 

74.05 

74.47 

74.90 

75.33 

75.77 

76.20 

76.64 

77.07 

77.51 

43 

.3 

77.95 

78.39 

78.84 

79.28 

79.73 

80.18 

80.63 

81.08 

81.54 

81.99 

45 

.4 

82.45 

82.91 

83.37 

83.83 

84.29 

84.76 

85.23 

85.70 

86.17 

86.64 

47 

5.5 

87.11 

87.59 

88.07 

88.55 

89.03 

89.51 

90.00 

90.48 

90.97 

91.46 

48 

.6 

91.95 

92.45 

92.94 

93.44 

93.94 

94.44 

94.94 

95.44 

95.95 

96.46 

50 

.7 

96.97 

97.48 

97.99 

98.51 

99.02 

99.54 

10006 

52 

.7 

100.1 

100.6 

101.1 

101.6 

5 

.8 

102.2 

102.7 

103.2 

103.8 

104.3 

104.8 

105.4 

105.9 

106.4 

107.0 

5 

.9 

107.5 

108.1 

108.6 

109.2 

109.7 

110.3 

110.9 

111.4 

112.0 

112.5 

6 

6.0 

113.1 

113.7 

114.2 

114.8 

115.4 

115.9 

116.5 

117.1 

117.7 

118.3 

6 

.1 

118.8 

119.4 

120.0 

120.6 

121.2 

121.8 

122.4 

123.0 

123.6 

124.2 

.2 

124.8 

125.4 

126.0 

126.6 

127.2 

127.8 

128.4 

129.1 

129.7 

130.3 

.3 

130.9 

131.5 

132.2 

132.8 

133.4 

134.1 

134.7 

135.3 

136.0 

136.6 

.4 

137.3 

137.9 

138.5 

139.2 

139.8 

140.5 

141.2 

141.8 

142.5 

143.1 

7 

6.5 

143.8 

144.5 

145.1 

145.8 

146.5 

147.1 

147.8 

148.5 

149.2 

149.8 

.6 

150.5 

151.2 

151.9 

152.6 

153.3 

154.0 

154.7 

155.4 

156.1 

156.8 

.7 

157.5 

158.2 

158.9 

159.6 

160.3 

161.0 

161.7 

162.5 

163.2 

163.9 

.8 

164.6 

165.4 

166.1 

166.8 

167.6 

168.3 

169.0 

169.8 

170.5 

171.3 

.9 

172.0 

172.8 

173.5 

174.3 

175.0 

175.8 

176.5 

177.3 

178.1 

178.8 

8 

7.0 

179.6 

180.4 

181.1 

181.9 

182.7 

183.5 

184.3 

185.0 

185.8 

186.6 

.1 

187.4 

188.2 

189.0 

189.8 

190.6 

191.4 

192.2 

193.0 

193.8 

194.6 

.2 

195.4 

196.2 

197.1 

197.9 

198.7 

199.5 

200.4 

201.2 

202.0 

202.9 

.3 

203.7 

204.5 

205.4 

206.2 

207.1 

207.9 

208.8 

209.6 

210.5 

211.3 

.4 

212.2 

213.0 

213.9 

214.8 

215.6 

216.5 

217.4 

218.3 

219.1 

220.0 

9 

7.5 

220.9 

221.8 

222.7 

223.6 

224.4 

225.3 

226.2 

227.1 

228.0 

228.9 

.6 

229.8 

230.8 

231.7 

232.6  - 

233.5 

234.4 

235.3 

236.3 

237.2 

238.1 

.7 

239.0 

240.0 

240.9 

241.8 

242.8 

243.7 

244.7 

245.6 

246.6 

247.5 

.8 

248.5 

249.4 

250.4 

251.4 

252.3 

253.3 

254.3 

255.2 

256.2 

257.2 

10 

.9 

258.2 

259.1 

260.1 

261.1 

262.1 

263.1 

264.1 

265.1 

266.1 

267.1 

8.0 

268.1 

269.1 

270.1 

271.1 

272.1 

273.1 

274.2 

275.2 

276.2 

277.2 

.1 

278.3 

279.3 

280.3 

281.4 

282.4 

283.4 

284.5 

285.5 

286.6 

287.6 

.2 

288.7 

289.8 

290.8 

291.9 

292.9 

294.0 

295.1 

296.2 

297.2 

298.3 

II 

3 

299.4 

300.5 

301.6 

302.6 

303.7 

304.8 

305.9 

307.0 

308.1 

309.2 

.4 

310.3 

311.4 

312.6 

313.7 

314.8 

315.9 

317.0 

318.2 

319.3 

320.4 

8.5 

321.6 

322.7 

323.8 

325.0 

326.1 

327.3 

328.4 

329.6 

330.7 

331.9 

.6 

333.0 

334.2 

335.4 

336.5 

337.7 

338.9 

340.1 

341.2 

342.4 

343.6 

12 

.7 

344.8 

346.0 

347.2 

348.4 

349.6 

350.8 

352.0 

353.2 

354.4 

355.6 

.8 

356.8 

358.0 

359.3 

360.5 

361.7 

362.9 

364.2 

365.4 

366.6 

367.9 

.9 

1 

369.1 

370.4 

371.6 

372.9 

374.1 

375.4 

376.6 

377.9 

379.2 

380.4 

13 

9.0 

381.7 

383.0 

384.3 

385.5 

386.8 

388.1 

389.4 

390.7 

392.0 

393.3 

.1 

394.6 

395.9 

397.2 

398.5 

399.8 

401.1 

402.4 

403.7 

405.1 

406.4 

2 

407.7 

409.1 

410.4 

411.7 

413.1 

414.4 

415.7 

417.1 

418.4 

419.8 

.3 

421.2 

422.5 

423.9 

425.2 

426.6 

428.0 

429.4 

430.7 

432.1 

433.5 

14 

.4 

434.9 

436.3 

437.7 

439.1 

440.5 

441.9 

443.3 

444.7 

446.1 

447.5 

9.5 

448.9 

450.3 

451.8 

453.2 

454.6 

456.0 

457.5 

458.9 

460.4 

461.8 

.6 

463.2 

464.7 

466.1 

467.6 

469.1 

470.5 

472.0 

473.5 

474.9 

476.4 

15 

.7 

477.9 

479.4 

480.8 

482.3 

483.8 

485.3 

486.8 

488.3 

489.8 

491.3 

.8 

492.8 

494.3 

495.8 

497.3 

498.9 

500.4 

501.9 

503.4 

505.0 

506.5 

16 

9 

508.0 

509.6 

511.1 

512.7 

514.2 

515.8 

517.3 

518.9 

520.5 

522.0 

10.0 

523.6 

Moving  the  decimal  point  ONE  place  in  D  requires  moving  it  THREE  places  in  body  of 
table  (see  p.  36). 


38 


MATHEMATICAL  TABLES 


SEGMENTS  OF  SPHERES 


(h  =  height  of  segment;  D  =  diam.  of  sphere) 


h 

Vol.  segm. 

to 

Q 

Vol.  segm. 

d 
S 

Explanation  of  Table  on  this  page 
Given,  h  =  height  of  segment, 
D  =  diam.  of  sphere. 

D 

D» 

Vol.  sphere 

0.00 
1 

0.0000 
0  0002 

2 

0.0000 
0  0003 

3 

2 

0.0006 

4 

0.0012 

9 

To  find  the  volume   of  the  segment, 

3 

0.0014 

10 

0.0026 

21 

form  the  ratio  h/D  and  find  from  the 

4 

0.0024 

14 

0.0047 

/  1 
26, 

table  the  value  of  (vol./D8);  then,  by 

0.05 

0.0038 

1  f 

0.0073 

of 

a  simple  multiplication, 

6 

g 

0.0054 
0.0073 
0  0095 

lo 
19 
22 

0.0104 
0.0140 
0  0182 

31 

36 
42 

vol.  segment  =  D*  X  (vo\./D*) 
The  table  gives  also  the  ratio  of  the 

9 

0.0120 

25 

*>7 

0.0228 

46 

11 

volume  of  the  segment  to  the  entire 

0.10 
1 

0.0147 
0  0176 

LI 

29 

0.0280 
0  0336 

_>/ 
56 

volume  of  the  sphere. 
NOTE.     Area  of  zone  =  v  X  h  X  D. 

2 

0.0208 

32 

0  A 

0.0397 

61 

// 

(Use  Table  of  Multiples  of  ir,  p.  28) 

3 

0  0242 

34 

0  0463 

oo 

4 

0.0279 

37 
39 

0.0533 

70 

74 

Explanation  of  Table  on  p.  34 

0.15 
6 

0.0318 
0  0359 

41 

0.0607 
0  0686 

79 

Given,  h  =  height  of  segment, 
c  =  chord. 

7 

0.0403 

44 

AC 

0.0769 

83 

QA 

To   find   the   diam.  of    the    circle,  the 

8 
9 

0.0448 
0.0495 

4-> 

47 
50 

0.0855 
0.0946 

OO 

91 
94 

length  of  arc,  or  the  area  of  the  seg- 
ment,   form   the   ratio   h/c,   and   find 

0.20 

0.0545 

C  1 

0.1040 

QO 

from  the  table  the  value  of  (diam./c), 

1 
2 
3 

0.05% 
0.0649 
0  0704 

>] 

53 
55 

0.1138 
0.1239 
0  1344 

Vo 

101 
105 

(arc/c),  or  (area/Ac)  ;  then,  by  a  simple 
multiplication, 

4 

0.0760 

56 
58 

0.1452 

108 
110 

diam.         =  c  X  (diam./c), 

arc             =  c  X  (arc/c), 

0.25 
6 

0.0818 
0  0878 

60 

0.1562 
0  1676 

114 

area           =  h  X  c  X  (area/Ac). 

7 

0.0939 

61 

0.1793 

117 

i  on 

The   table   gives   also   the   angle   sub- 

8 

0.1002 

63 

64 

0.1913 

120 

122 

tended  at  the  center,  and  the  ratio  of 

9 

0.1066 

65 

0.2035 

125 

h  to  D.     See  p.  106. 

0.30 
1 

0.1131 
0  1198 

67 

0.2160 
0  2287 

127 

Explanation  of  Table  on  p.  35 

2 

0.1265 

67 

Xrt 

0.2417 

130 

Given,  h  =  height  of  segment, 

3 

0.1334 

ov 
70 

0.2548 

134 

D  —  diam.  of  circle. 

4 

0.1404 

/u 
71 

0.2682 

135 

To  find  the  chord,  the  length  of  arc, 

0.35 

0.1475 

n«i 

0.2817 

1  Ifl 

or  the  area  of  the  segment,  form  the 

6 
7 
8 

0.1547 
0.1620 
0.1694 

n 

73 
74 

0.2955 
0.3094 
0.3235 

1  JO 

139 
141 

ratio  h/D,  and  find  from  the  table  the 
value     of     (chord/D),      (arc/Z>),     or 

9 

0.1768 

74 
75 

0.3377 

142 
143 

(area//)2);    then,  by  a   simple   multi- 

MJ 

plication, 

0.40 

0.1843 
0.1919 

76 

0.3520 
0.3665 

145 

chord  =  D  X  (chord/  D), 

2 

0.1995 

76 

77 

0.3810 

145 

147 

arc       -  D  X  (arc//)), 

3 
4 

0.2072 
0.2149 

II 

77 
78 

0.3957 
0.4104 

It/ 

147 
148 

area     =  D*  X  (area/Z)"). 
The   table    gives   also   the    angle   sub- 

0.45 

0.2227 

•70 

0.4252 

1  AQ 

tended  at  the  center,  the  ratio  of  the 

6 
7 
8 

0.2305 
0.2383 
0  2461 

7o 
78 
78 

0.4401 
0.4551 
0  4700 

1^7 

150 
149 

arc  of  the  segment  to  the  whole  cir- 
cumference, and  the  ratio  of  the  area 

9 

0.2539 

78 
79 

.  0.4850 

150 
150 

of   the   segment   to   the    area   of   the 

whole  circle.     See  p.  106. 

0.50 

0.2618 

0.5000 

NOTE.     Vol.  segm.  -  }6  *  h*  (3D-2h). 


MATHEMATICAL  TABLES 


39 


REGULAR  POLYGONS 

n  =•  number  of  sides; 

TO  =  360°/n  =  angle  subtended  at  the  center  by  one  side; 

a  =  length  of  one  side  =  R  (2  sin  |-)   =  r  (2  tan  |-)  ; 
R  =  radius  of  circumscribed  circle  =  a  (  y^  esc  —  \  =  r  (sec  -^-j ; 
r  =  radius  of  inscribed  circle  =  R(COS  —  J   =  a(l$  cot  s~J  ; 
Area  =  o*H  n  cot    -     =  fl'/i  n  sin  t>     =  r*n  tan    -. 


Area 

Area 

Area 

* 

R 

a 

a 

r 

r 

n 

7) 

a2 

#» 

r2 

0 

r 

R 

r 

R 

a 

3 

120° 

0.4330 

1.299 

5:196 

0.5774 

2.000 

1.732 

3.464 

0.5000 

0.2887 

4 

90° 

1.000 

2.000 

4.000 

0.7071 

.414 

1.414 

2.000 

0.7071 

0.5000 

5 

72° 

1.721 

2.378 

3.633 

0.8507 

.236 

1.176 

1.453 

0.8090 

0.6882 

6  . 

60° 

2.598 

2.598 

3.464 

1.0000 

.155 

1.000 

1.155 

0.8660 

0.8660 

7 

5P.43 

3.634 

2.736 

3.371 

1.152 

.110 

0.8678 

0.9631 

0.9010 

1.038 

8 

45° 

4.828 

2.828 

3.314 

1.307 

.082 

0.7654 

0.8284 

0.9239 

1.207 

9 

40° 

6.182 

2.893 

3.276 

1.462 

.064 

0.6840 

0.7279 

0.9397 

1.374 

10 

36° 

7.694 

2.939 

3.249 

1.618 

.052 

0.6180 

0.6498 

0.9511 

1.539 

12 

30° 

11.20 

3.000 

3.215 

1.932 

.035 

0.5176 

0.5359 

0.9659 

1.866 

15 

24° 

17.64 

3.051 

3.188 

2.405 

.022 

0.4158 

0.4251 

0.9781 

2.352 

16 

22°.  50 

20.11 

3.062 

3.183 

2.563 

.020 

0.3902 

0.3978 

0.9808 

2.514 

20 

18° 

31.57 

3.090 

3.168 

3.1% 

.013 

0.3129 

0.3168 

0.9877 

3.157 

24 

15° 

45.58 

3.106 

3.160 

3.831 

.009 

0.2611 

0.2633 

0.9914 

3.798 

32 

11°.25 

81.23 

3.121 

3.152 

5.101 

.005 

0.1960 

0.1970 

0.9952 

5.077 

48 

7°.  50 

183.1 

3.133 

3.146 

7.645 

.002 

0.1308 

0.1311 

0.9979 

7.629 

64 

5°.625 

325.7 

3.137 

3.144 

10.19 

.001 

0.0981 

0.0983 

0.9988 

10.18 

BINOMIAL  COEFFICIENTS 

(For  table  giving  binomial  coefficients  for  fractional  values  of  n,  see  p.  116). 
n(n  -  1)  n(n  -  l)(n  -  2) 


(n)o  =  1;  (n)i  =  n;  (71)2  = 

(n) 


1X2' 

n(n  -  l)(n  -  2)   .    .    .    (n  -  [r  -  1]). 
1X2X3.    .    .    X  r 


1X2X3 
Another  notation: 


=  (n)r. 


n 

(n)o 

(»)i 

(n)i 

(n)i 

(n)* 

(n). 

(n)a 

(»)T 

(n)a 

(n)a 

(n)w 

(n)u 

(n)is 

(n)i3 

I 

1 

? 

2 

1 

3 

3 

3 

1 

4 

4 

6 

4 

1 

5 

5 

10 

10 

5 

1 

6 

6 

15 

20 

15 

6 

1 

7 

7 

21 

35 

35 

71 

7 

1 

8 

8 

28 

56 

70 

56 

28 

8 

1 

Q 

9 

36 

84 

126 

126 

84 

36 

9 

1 

in 

10 

45 

120 

210 

252 

210 

120 

45 

10 

| 

11 

11 

51 

165 

310 

462 

462 

330 

165 

55 

11 

1 

12 
13 
14 
15 

12 
13 
14 
15 

66 
78 
91 
105 

220 
286 
364 
455 

495 
715 
1001 
1365 

792 
1287 
2002 
3003 

924 
1716 
3003 
5005 

792 
1716 
3432 
6435 

495 
1287 
3003 
6435 

220 
715 
2002 
5005 

66 
286 
1001 
3003 

12 
78 
364 
1365 

1 

13 
91 
455 

..... 

14 
105 

For  n  =  14,  (n)u  =  1;  for  n  -  15,  (n)u  =  15,  and  (n)l6  -  1. 


40  MATHEMATICAL  TABLES 

COMMON  LOGARITHMS    (special  table) 


1* 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

$?»• 

«JTJ 

1.00 

0.0000 

0004 

0009 

0013 

0017 

0022 

0026 

0030 

0035 

0039 

4 

1.01 

0043 

0048 

0052 

0056 

0060 

0065 

0069 

0073 

0077 

0082 

1.02 

0086 

0090 

0095 

0099 

0103 

0107 

0111 

0116 

0120 

0124 

1.03 

0128 

0133 

0137 

0141 

0145 

0149 

0154 

0158 

0162 

0166 

1.04 

0170 

0175 

0179 

0183 

0187 

0191 

0195 

0199 

0204 

0208 

1.05 

0212 

0216 

0220 

0224 

0228 

0233 

0237 

0241 

0245 

0249 

1.06 

0253 

0257 

0261 

0265 

0269 

0273 

0278 

0282 

0286 

0290 

1.07 

0294 

0298 

0302 

0306 

0310 

0314 

0318 

0322 

0326 

0330 

.08 

0334 

0338 

0342 

0346 

0350 

0354 

0358 

0362 

0366 

0370 

1.09 

0374 

0378 

0382 

0386 

0390 

0394 

0398 

0402 

04C6 

0410 

.10 

0.0414 

0418 

0422 

0426 

0430 

0434 

0438 

0441 

0445 

0449 

.11 

0453 

0457 

0461 

0465 

0469 

0473   . 

0477 

0481 

0484 

0488 

.12 

0492 

0496 

0500 

0504 

0508 

0512 

0515 

0519 

0523 

0527 

.13 

0531 

0535 

0538 

0542 

0546 

0550 

0554 

0558 

0561 

0565 

.14 

0569 

0573 

0577 

0580 

0584 

0588 

0592 

0596 

0599 

0603 

.15 

0607 

0611 

0615 

0618 

0622 

0626 

0630 

0633 

0637 

0641 

.16 

0645 

0648 

0652 

0656 

0660 

0663 

0667 

0671 

0674 

0678 

.17 

0682 

0686 

0689 

0693 

0697 

0700 

0704 

0708 

0711 

0715 

.18 

0719 

0722 

0726 

0730 

0734 

0737 

0741 

0745 

0748 

0752 

.19 

0755 

0759 

0763 

0766 

0770 

0774 

0777 

0781 

0785 

0788 

x  .• 

1.20 

0.0792 

0795 

0799 

0303 

0806 

0810 

0813 

0817 

0821 

0824 

1.21 

0828 

0831 

0835 

0839 

0842 

0846 

0849 

0853 

0856 

0860 

1.22 

0864 

0867 

0871 

0374 

0878 

0881 

0885 

0888 

0892 

0896 

1.23 

0899 

0903 

0906 

0910 

0913 

0917 

0920 

0924 

0927 

0931 

1.24 

0934 

0938 

0941 

0945 

0948 

0952 

0955 

0959 

0962 

0966 

1.25 

0969 

0973 

0976 

0980 

0983 

0986 

0990 

0993 

0997 

1000 

3 

1.26 

1004 

1007 

1011 

1014 

1017 

1021 

1024 

1028 

1031 

1035 

1.27 

1038 

1041 

1045 

1048 

1052 

1055 

1059 

1062 

1065 

1069 

1.28 

1072 

1075 

1079 

1082 

1086 

1089 

1092 

1096 

1099 

1103 

1.29 

1106 

1109 

1113 

1116 

1119 

1.123 

1126 

1129 

1133 

1136 

1.30 

0.1139 

1143 

1146 

1149 

1153 

1156 

1159 

1163 

1166 

1169 

1.31 

1173 

1176 

1179 

1183 

1186 

1189 

1193 

1196 

1199 

1202 

1.32 

1206 

1209 

1212 

1216 

1219 

1222 

1225 

1229 

1232 

1235 

1.33 

1239 

1242 

1245 

1248 

1252 

1255 

1258 

1261 

1265 

1268 

1.34 

1271 

1274 

1278 

1281 

1284 

1287 

1290 

1294 

1297 

1300 

1.35 

1303 

1307 

1310 

1313 

1316 

1319 

1323 

1326 

1329 

1332 

1.36 

1335 

1339 

1342 

1345 

1348 

1351 

1355 

1358 

1361 

1364 

1.37 

1367 

1370 

1374 

1377 

1380 

1383 

1386 

1389 

1392 

1396 

1.38 

1399 

1402 

1405 

1408 

1411 

1414 

1418 

1421 

1424 

1427 

1.39 

1430 

1433 

1436 

1440 

1443 

1446 

1449 

1452 

1455 

1458 

1.40 

0.1461 

1464 

1467 

1471 

1474 

1477 

1480 

1483 

1486 

1489 

1.41 

1492 

1495 

1498 

1501 

1504 

1508 

1511 

1514 

1517 

1520 

1.42 

1523 

1526 

1529 

1532 

1535 

1538 

1541 

1544 

1547 

1550 

1.43 

1553 

1556 

1559 

1562 

1565 

1569 

1572 

1575 

1578 

1581 

1.44 

1584 

1587 

1590 

1593 

1596 

1599 

1602 

1605 

1608 

1611 

1.45 

1614 

1617 

1620 

1623 

1626 

1629 

1632 

1635 

1638 

1641 

1.46 

1644 

1647 

1649 

1652 

1655 

1658 

1661 

1664 

1667 

1670 

1.47 

1673 

1676 

1679 

1682 

1685 

1688 

1691 

1694 

1697 

1700 

1.48 

1703 

1706 

1708 

1711 

1714 

1717 

1720 

1723 

1726 

1729 

1.49 

1732 

1735 

1738 

1741 

1744 

1746 

1749 

1752 

1755 

1758 

Moving  the  decimal  point  n  places  to  the  right  [or  left]  in  the  number  requires  adding  +  n 
[or  -  n]  ia  the  body  of  the  table  (see  p.  42). 


MATHEMATICAL  TABLES 


41 


COMMON  LOGARITHMS   (special  table,  continued) 


p 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

11 

1.50 

0.1761 

1764 

1767 

1770 

1772 

1775  . 

1778 

1781 

1784 

1787 

3 

1.51 

1790 

1793 

1796 

1798 

1801 

1804 

1807 

1810 

1813 

1816 

1.52 

1818 

1821 

1824 

1827 

1830 

1833 

1836 

1838 

1841 

1844 

1.53 

1847 

1850 

1853 

1855 

1858 

1861 

1864 

1867 

1870 

1872 

1.54 

1875 

1878 

1881 

1884 

1886 

1889 

1892 

1895 

1898 

1901 

1.55 

1903 

1906 

1909 

1912 

19t5 

1917 

1920 

1923 

1926 

1928 

1.56 

1931 

1934 

1937 

1940 

1942 

1945 

1948 

1951 

1953 

1956 

1.57 

1959 

1962 

1965 

1967 

1970 

1973 

1976 

1978 

1981 

1984 

1.58 

1987 

1989 

1992 

1995 

1998 

2000 

2003 

2006 

2009 

2011 

1.59 

2014 

2017 

2019 

2022 

2025 

2028 

2030 

2033 

2036 

2038 

1.60 

0.2041 

2044 

2047 

2049 

2052 

2055 

2057 

2060 

2063 

2066 

1.61 

2068 

2071 

2074 

2076 

2079 

2082 

2084 

2087 

2090 

2092 

1.62 

•  2095 

2098 

2101 

2103 

2106 

2109 

2111 

2114 

2117 

2119 

1.63 

2122 

2125 

2127 

2130 

2133 

2135 

2138 

2140 

2143 

2146 

1.64 

2148 

2151 

2154 

2156 

2159 

2162 

2164 

2167 

2170 

2172 

1.65 

2175 

2177 

2180 

2183 

2185 

2188 

2191 

2193 

2196 

2198 

1.66 

2201 

2204 

2206 

2209 

2212 

2214 

2217 

2219 

2222 

2225 

1.67 

2227 

2230 

2232 

2235 

2238 

2240 

2243 

2245 

2248 

2251 

1.68 

2253 

2256 

2258 

2261 

2263 

2266 

2269 

2271 

2274 

2276 

1.69 

2279 

2281 

2284 

2287 

2289 

2292 

2294 

2297 

2299 

2302 

1.70 

0.2304 

2307 

2310 

2312 

2315 

2317 

2320 

2322 

2325 

2327 

1.71 

2330 

2333 

2335 

2338 

2340 

2343 

2345 

2348 

2350 

2353 

1.72 

2355 

2358 

2360 

2363 

2365 

2368 

2370 

2373 

2375 

2378 

1.73 

2380 

2383 

2385 

2388 

2390 

2393 

2395 

2398 

2400 

2403 

1.74 

2405 

2408 

2410 

2413 

2415 

2418 

2420 

2423 

2425 

2428 

2 

1.75 

2430 

2433 

2435 

2438 

2440 

2443 

2445 

2448 

2450 

2453 

1.76 

2455 

2458 

2460 

2463 

2465 

2467 

2470 

2472 

2475 

2477 

1.77 

2480 

2482 

2485 

2487 

2490 

2492 

2494 

2497 

2499 

2502 

1.78- 

2504 

2507 

2509 

2512 

2514 

2516 

•  2519 

2521 

2524 

2526 

1.79 

2529 

2531 

2533 

2536 

2538 

2541 

2543 

2545 

2548 

2550 

180 

0.2553 

2555 

2558 

2560 

2562 

2565 

2567 

2570 

2572 

2574 

1.81 

2577 

2579 

2582 

2584 

2586 

2589 

2591 

2594 

2596 

2598 

1.82 

2601 

2603 

2605 

2608 

2610 

2613 

2615 

2617 

2620 

2622 

1.83 

2625 

2627 

2629 

2632 

2634 

2636 

2639 

2641 

2643 

2646 

1.84 

2648 

2651 

2653 

2655 

2658 

2660 

2662 

2665 

2667 

2669 

1.85 

2672 

2674 

2676 

2679 

2681 

2683 

2686 

2688 

2690 

2693 

1.86 

2695 

2697 

2700 

2702 

2704 

2707 

2709 

2711 

2714 

2716 

1.87 

2718 

2721 

2723 

2725 

2728 

2730 

2732 

2735 

2737 

2739 

1.88 

2742 

2744 

2746 

2749 

2751 

2753 

2755 

2758 

2760 

2762 

1.89 

2765 

2767 

2769 

2772 

2774 

2776 

2778 

2781 

2783 

2785 

1.90 

0.2788 

2790 

2792 

2794 

2797 

2799 

2801 

2804 

2806 

2808 

1.91 

2810 

2813 

2815 

2817 

2819 

2822 

2824  ' 

2826 

2828 

2831 

1.92 

2833 

2835 

2838 

2840 

2842 

2844 

2847 

2849 

2851 

2853 

1.93 

2856 

2858 

2860 

2862 

2865 

2867 

2869 

2871 

2874 

2876 

1.94 

2878 

2880 

2882 

2885 

2887 

2889 

2891 

2894 

2896 

2898 

1.95 

2900 

2903 

2905 

2907 

2909 

2911 

2914 

2916 

2918 

2920 

1.96 

2923 

2925 

2927 

2929 

2931 

2934 

2936 

2938 

2940 

2942 

1.97 

2945 

2947 

2949 

2951 

2953 

2956 

2958 

2960 

2962 

2964 

1.98 

2967 

2969 

2971 

2973 

2975 

2978 

2980 

2982 

2984 

2986 

1.99 

2989 

2991 

2993 

2995 

2997 

2999 

3002 

3004 

3006 

3008 

42 


MATHEMATICAL  TABLES 


COMMON  LOGARITHMS 


11 

0 

1 

2 

a 

4 

5 

6 

7 

8 

9 

Sa 

<3-0 

1.0 

0.0000 

0043 

0086 

0128 

0170 

0212 

0253 

0294 

0334 

0374 

1.1 

0414 

0453 

0492 

0531 

0569 

0607 

0645 

0682 

0719 

0755 

1.2 

0792 

0828 

0864 

0899 

0934 

0969 

1004 

1038 

1072 

1106 

TH 

1.3 

1139 

1173 

1206 

1239 

1271 

1303 

1335 

1367 

1399 

1430 

1 

1.4 

1461 

1492 

1523 

1553 

1584 

1614 

1644 

1673 

1703 

1732 

6 

•^ 

13 

1761 

1790 

1818 

1847 

1875 

1903 

1931 

1959 

1987 

2014 

1 

1.6 

2041 

2068 

2095 

2122 

2148 

2175 

2201 

2227 

2253 

2279 

03 

1.7 

2304 

2330 

2355 

2380 

2405 

2430 

2455 

2480 

2504 

2529 

ft 

1.8 

2553 

2577 

2601 

2625 

2648 

2672 

2695 

2718 

2742 

2765 

1 

1.9 

2788 

2810 

2833 

2856 

2878 

2900 

2923 

2945 

2967 

2989 

CQ 

2.0 

0.3010 

3032 

3054 

3075 

3096 

3118 

3139 

3160 

3181 

3201 

21 

2.1 

3222 

3243 

3263 

3284 

3304 

3324 

3345 

3365 

3385 

3404 

20 

2.2 

3424 

3444 

3464 

3483 

3502 

3522 

3541 

3560 

3579 

3598 

19 

2.3 

3617 

3636 

3655 

3674 

3692 

3711 

3729 

3747 

3766 

3784 

18 

2.4 

3802 

3820 

3838 

3856 

3874 

3892 

3909 

3927 

3945 

3962 

17 

2.5 

3979 

3997 

4014 

4031 

4048 

4065 

4082 

4099 

4116 

4133 

17 

2.6 

4150 

4166 

4183 

4200 

4216 

4232 

4249 

4265 

4281 

4298 

16 

2.7 

4314 

4330 

4346 

4362 

4378 

4393 

4409 

4425 

4440 

4456 

16 

2.8 

4472 

4487 

4502 

4518 

4533 

4548 

4564 

4579 

4594 

4609 

15 

2.9 

4624 

4639 

4654 

4669 

4683 

4698 

4713 

4728 

4742 

4757 

15 

3.0 

0.4771 

4786 

4800 

4814 

4829 

4843 

4857 

4871 

4886 

4900 

14 

3.1 

4914 

4928 

4942 

4955 

4969 

4983 

4997 

5011 

5024 

5038 

14 

3.2 

5051 

5065 

5079 

5092 

5105 

5119 

5132 

5145 

5159 

5172 

13 

3.3 

5185 

5198 

5211 

5224 

5237 

5250 

5263 

5276 

5289 

5302 

13 

3.4 

5315 

5328 

5340 

5353 

5366 

5378 

5391 

5403 

5416 

5428 

13 

3.5 

5441 

5453- 

5465 

5478 

5490 

5502 

5514 

5527 

5539 

5551 

12 

3.6 

5563 

5575 

5587 

5599 

5611 

5623 

5635 

5647 

5658 

5670 

12 

3.7 

5682 

5694 

5705 

5717 

5729 

5740 

5752 

5763 

5775 

5786 

12 

3.8 

5798 

5809 

5821 

5832 

5843 

5855 

5866 

5877 

5888 

5899 

11 

3.9 

5911 

5922 

5933 

5944 

5955 

5966 

5977 

5988 

5999 

6010 

11 

4.0 

0.6021 

6031 

6042 

6053 

6064 

6075 

6085 

6096 

6107 

6117 

11 

4.1 

6128 

6138 

6149 

6160 

6170 

6180 

6191 

6201 

6212 

6222 

10 

4.2 

6232 

6243 

6253 

6263 

6274 

6284 

6294 

6304 

6314 

6325 

10 

4.3 

6335 

6345 

6355 

6365 

6375 

6385 

6395 

6405 

6415 

6425 

10 

4.4 

6435 

6444 

6454 

6464 

6474 

6484 

6493 

6503 

6513 

6522 

10 

4.5 

6532 

6542 

6551 

6561 

6571 

6580 

6590 

6599 

6609 

6618 

10 

4.6 

6628 

6637 

6646 

6656 

6665 

6675 

6684 

6693 

6702 

6712 

10 

4.7 

6721 

6730 

6739 

6749 

6758 

6767 

6776 

6785 

6794 

6803 

9 

4.8 

6812 

6821 

6830 

6839 

6848 

6857 

6866 

6875 

6884 

6893 

9 

4.9 

6902 

6911 

6920 

6928 

6937 

6946 

6955 

6964 

6972 

6981 

9 

log  TT  =  0.4971 
log  e  =  0.4343 


log  7T/2  =  0.1961         log  71-2 
log  (0.4343)  =  0.6378  -  1 


0.9943 


log 


0.2486 


These  two  pages  give  the  common  logarithms  of  numbers  between  1  and  10,  correct 
to  four  places.     Moving  the  decimal  point  n  places  to  the  right  [or  left]  in  the  number  is 
equivalent  to  adding  n[dr-n]  to   the  logarithm.     Thus,  log  0.017453  =  0.2419  -  2, 
which  may  also  be  written  2.2419  or  8.2419  -  10.     See  p.  91.     Graphs,  p.  174. 
log  (aft)  =  log  a  +  log  6  log  (aN)  =  N  log  a 


log  =  log  a  -  log  6 


log 


-       log  a 


MATHEMATICAL  TABLES 


43 


COMMON  LOGARITHMS  (continued) 


1* 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

!i 

5.0 

0.6990 

6998 

7007 

7016 

7024 

7033 

7042 

7050 

7059 

7067 

9 

5.1 

7076 

7084 

7093 

7101 

7110 

7118 

7126 

7135 

7143 

7152 

8 

5.2 

7160 

7168 

7177 

7185 

7193 

7202 

7210 

7218 

7226 

7235 

8 

5.3 

7243 

7251 

7259 

7267 

7275 

7284 

7292 

7300 

7308 

7316 

8 

5.4 

7324 

,   7332 

7340 

7348 

7356 

7364 

7372 

7380 

7388 

7396 

8 

5.5 

7404 

7412 

7419 

7427 

7435 

7443 

7451 

7459 

7466 

7474 

8 

5.6 

7482 

7490 

7497 

7505 

7513 

7520 

7528 

7536 

7543 

7551 

8 

5.7 

7559 

>  7566 

7574 

7582 

7589 

7597 

7604 

7612 

7619 

7627 

8 

5.8 

7634 

7642 

7649 

7657 

7664 

7672 

7679 

7686 

7694 

7701 

7 

5.9 

7709 

7716 

7723 

7731 

7738 

7745 

7752 

7760 

7767 

7774 

7 

6.0 

0.7782 

7789 

7796 

7803 

7810 

7818 

7825 

7832 

7839 

7846 

7 

6.1 

7853 

7860 

7868 

7875 

7882 

7889 

7896 

7903 

7910 

7917 

7 

6.2 

7924 

7931 

7938 

7945 

7952 

7959 

7966 

7973 

7980 

7987 

7 

6.3 

7993 

8000 

8007 

8014 

8021 

8028 

8035 

8041 

8048 

8055 

7 

6.4 

8062 

8069 

8075 

8082 

8089 

8096 

.  8102 

8109 

8116 

8122 

7 

6.5 

8129 

8136 

8142 

8149 

8156 

8162 

8169 

8176 

8182 

8189 

7 

6.6 

8195 

8202 

8209 

8215 

8222 

8228 

8235 

8241 

8248 

8254 

7 

6.7 

8261 

8267 

8274 

8280 

8287 

8293 

8299 

8306 

8312 

8319 

6 

6.8 

8325 

8331 

8338 

8344 

8351 

8357 

8363 

8370 

8376 

8382 

6 

6.9 

8388 

8395 

8401 

8407 

8414 

8420 

8426 

8432 

8439 

8445 

6 

7.0 

0.8451 

8457 

8463 

8470 

8476 

8482 

8488 

8494 

8500 

8506 

6 

7.1 

1  8513 

8519 

8525 

8531 

8537 

8543 

8549 

8555 

8561 

8567 

6 

7.2 

8573 

8579 

8585 

8591 

8597 

8603 

8609 

8615 

8621 

8627 

6 

7.3 

8633 

8639 

8645 

8651 

8657 

8663 

8669 

8675 

8681 

8686 

6 

7.4 

8692 

8698 

8704 

8710 

8716 

8722 

8727 

8733 

8739 

8745 

6 

7.5 

8751 

8756 

8762 

8768 

8774 

8779 

8785 

8791 

8797 

8802 

6 

7.6 

8808 

8814 

8820 

8825 

8831 

8837 

8842 

8848 

8854 

8859 

6 

7.7 

8865 

8871 

8876 

8882 

8887 

8893 

8899 

8904 

8910 

8915 

6 

7.8 

8921 

8927 

8932 

8938 

8943 

8949 

8954 

8960 

8965 

8971 

6 

7.9 

8976 

8982 

8987 

8993 

8998 

9004 

9009 

9015 

9020 

9025 

5 

8.0 

0.9031 

9036 

9042 

9047 

9053 

9058 

9063 

9069 

9074 

9079 

5 

8.1 

9085 

9090 

9096 

9101 

9106 

9112 

9117 

9122 

9128 

9133 

5 

8.2 

9138 

9143 

9149 

9154 

9159 

9165 

9170 

9175 

9180 

9186 

5 

8.3 

9191 

9196 

9201 

9206 

9212 

9217 

9222 

9227 

9232 

9238 

5 

8.4 

9243 

9248 

9253 

9258 

9263  . 

9269 

9274 

9279 

9284 

9289 

5 

8.5 

9294 

9299 

9304 

9309 

9315 

9320 

9325 

9330 

9335 

9340 

5 

8.6 

9345 

9350 

9355 

9360 

9365 

9370 

9375 

9380 

9385 

9390 

5 

8.7 

9395 

9400 

9405 

9410 

9415 

9420 

9425 

9430 

9435 

9440 

5 

8.8 

9445 

9450 

9455 

9460 

9465 

9469 

9474 

9479 

9484 

9489 

5 

8.9 

9494 

.  9499 

9504 

9509 

9513 

9518 

9523 

9528 

9533 

9538 

5 

9.0 

0.9542 

9547 

9552 

9557 

9562 

9566 

9571 

9576 

9581 

9586 

5 

9.1 

9590 

9595 

9600 

9605 

9609 

9614 

9619 

9624 

9628 

9633 

5 

9.2 

9638 

9643 

9647 

9652 

9657 

9661 

9666 

9671 

9675 

9680 

5 

9.3 

9685 

9689 

9694 

9699 

9703 

9708 

9713 

9717 

9722 

9727 

5 

9.4 

9731 

9736 

9741 

9745 

9750  , 

9754 

9759 

9763 

9768 

9773 

5 

9.5 

9777 

9782 

9786 

9791 

9795 

9800 

9805 

9809 

9814 

9818 

5 

9.6 

9823 

9827 

9832 

9836 

9841 

9845 

9850 

9854 

9859 

9863 

4 

9.7 

9868 

9872 

9877 

9881 

9886 

9890 

9894 

9899 

9903 

9908 

4 

9.8 

99  r2- 

-   9917 

9921 

9926 

9930 

9934 

9939 

9943 

9948 

9952 

4 

9.9 

9956 

9961 

9965 

9969 

9974 

9978 

9983 

9987 

9991 

9996 

4 

44 


MATHEMATICAL  TABLES 


DEGREES  AND  MINUTES  EXPRESSED  IN  RADIANS  (See  also  p.  69) 


Degrees 

Hundredths 

Minutes 

1° 

.0175 

61° 

1.0647 

121° 

2.1118 

0°.01 

.0002 

0°.51 

.0089 

r 

.0003 

2 

.0349 

2 

1.0821 

2 

2.1293 

2 

.0003 

2 

.0091 

2' 

.0006 

3 

.0524 

3 

.0996 

3 

2.1468 

3 

.0005 

3 

.0093 

3' 

.0009 

4 

.0698 

4 

.1170 

4 

2.1642 

4 

.0007 

4 

.0094 

4' 

.0012 

5° 

.0873 

65° 

.1345 

125° 

2.1817 

.05 

.0009 

.55 

.0096 

5' 

.0015 

6 

.1047 

6 

.1519 

6 

2.1991 

6 

.0010 

6 

.0098 

6' 

.0017 

7 

.1222 

7 

.1694 

7 

2.2166 

7 

.0012 

7 

.0099 

7' 

.0020 

8 

.1396 

8 

.1868 

8 

2.2340 

8 

.0014 

8 

.0101 

8' 

.0023 

9 

.1571 

9 

1.2043 

9 

2.2515 

9 

.0016 

9 

.0103 

9' 

.0026 

10° 

.1745 

70° 

1.2217 

130° 

2.2689 

0°.10 

.0017 

0°.60 

.0105 

10' 

.0029 

.1920 

1 

1.2392 

1 

2.2864 

1 

.0019 

] 

.0106 

11' 

.0032 

2 

.2094 

2 

1.2566 

2 

2.3038 

2 

.0021 

2 

.0108 

12' 

.0035 

3 

.2269 

3 

1.2741 

3 

2.3213 

3 

.0023 

3 

.0110 

13' 

.0038 

4 

.2443 

4 

1.2915 

4 

2.3387 

4 

.0024 

4 

.0112 

14' 

.0041 

15° 

.2618 

75° 

1.3090 

136° 

2.3562 

.15 

.0026 

.65 

.0113 

15' 

0044 

6 

.2793 

6 

1.3265 

6 

2.3736 

6 

.0028 

6 

.0115 

16' 

.0047 

7 

.2967 

7 

1.3439 

7 

2.3911 

7 

.0030 

7 

.0117 

17' 

.0049 

8 

.3142 

8 

1,3614 

8 

2.4086 

8 

.0031 

8 

.0119 

18' 

.0052 

9 

.3316 

9 

1.3788 

9 

2.4260 

9 

.0033 

9 

.0120 

19' 

.0055 

20° 

.3491 

80° 

1.3963 

140° 

2.4435 

0°.20 

.0035 

0°.70 

.0122 

20' 

.0058 

1 

.3665 

1 

.4137 

1 

2.4609 

1 

.0037 

1 

.0124 

21' 

.0061 

2 

.3840 

2 

.4312 

2 

2.4784 

2 

.0038 

2 

.0126 

22' 

.0064 

3 

.4014 

3 

.4486 

3 

2.4958 

3 

.0040 

3 

.0127 

23' 

.0067 

4 

.4189 

4 

.4661 

4 

2.5133 

4 

.0042 

4 

.0129 

24' 

.0070 

25° 

.4363 

85° 

1.4835 

145° 

2.5307 

.25 

.0044 

.75 

.0131 

25' 

0073 

6 

.4538 

6 

1.5010 

6 

2.5482 

6 

.0045 

6 

.0133 

26' 

.0076 

7 

.4712 

7 

1.5184 

7 

2.5656 

7 

.0047 

7 

.0134 

27' 

.0079 

8 

.4887 

8 

1.5359 

8 

2.5831 

8 

.0049 

8 

.0136 

28' 

.0081 

9 

.5061 

9 

1.5533 

9 

2.6005 

9 

.0051 

9 

.0138 

29' 

0084 

30° 

.5236 

90° 

1.5708 

150° 

2.6180 

0°.30 

.0052 

0°.80 

.0140 

30' 

.0087 

1 

.5411 

1 

.5882 

1 

2.6354 

1 

.0054 

1 

.0141 

31' 

.0090 

2 

.5585 

2 

1.6057 

2 

2.6529 

2 

.0056 

2 

0143 

32' 

.0093 

3 

.5760 

3 

1.6232 

3 

2.6704 

3 

.0058 

3 

.0145 

33' 

.0096 

4 

.5934 

4 

1.6406 

4 

2.6878 

4 

.0059 

4 

.0147 

34' 

.0099 

35° 

.6109 

95° 

1.6581 

155° 

2.7053 

35 

.0061 

.85 

.0148 

35' 

.0102 

6 

.6283 

6 

1.6755 

6 

2.7227 

6 

.0063 

6 

.0150 

36' 

.0105 

7 

.6458 

7 

1.6930 

7 

2.7402 

7 

.0065 

7 

.0152 

37' 

.0108 

8 

.6632 

8 

1.7104 

8 

2.7576 

8 

.0066 

8 

.0154 

38' 

.0111 

9 

.6807 

9 

1.7279 

9 

2.7751 

9 

.0068 

9 

.0155 

39' 

.0113 

40° 

.6981 

100° 

1.7453 

160° 

2.7925 

0°.40 

.0070 

0°.90 

.0157 

40' 

.0116 

1 

.7156 

1 

1.7628 

1 

2.8100 

1 

.0072 

1 

.0159 

41' 

.0119 

2 

.7330 

2 

1.7802 

2 

2.8274 

2 

.0073 

2 

.0161 

42' 

.0122 

3 

.7505 

3 

1.7977 

3 

2.8449 

3 

.0075 

3 

.0162 

43' 

.0125 

4 

.7679 

4 

1.8151 

4 

2.8623 

4 

.0077 

4 

.0164 

44' 

.0128 

45° 

.7854 

105° 

1.8326 

165° 

2.8798 

.45 

.0079 

.95 

.0166 

45' 

.0131 

6 

.8029 

6 

1.8500 

6 

2.8972 

6 

.0080 

6 

.0168 

46' 

.0134 

7 

.8203 

7 

1.8675 

7 

2.9147 

7 

.0082 

7 

.0169 

47' 

.0137 

8 

.8378 

8 

1.8850 

8 

2.9322 

8 

.0084 

8 

.0171 

48' 

.0140 

9 

.8552 

9 

1.9024 

9 

2.9496 

9 

.0086 

9 

.0173 

49' 

.0143 

50° 

.8727 

110° 

1.9199 

170° 

2.9671 

0°.50 

.0087 

1°.00 

.0175 

50' 

.0145 

1 

.8901 

1 

1.9373 

1 

2.9845 

51' 

0148 

2 

.9076 

2 

1.9548 

2 

3.0020 

52' 

.0151 

3 

.9250 

3 

1.9722 

3 

3.0194 

53' 

0154 

4 

.9425 

4 

1.9897 

4 

3.0369 

54' 

.0157 

55° 

.9599 

115° 

2.0071 

175° 

3.0543 

55' 

.0160 

6 

.9774 

6 

2.0246 

6 

3.0718 

56' 

0163 

7 

.9948 

7 

2.0420 

7 

3.0892 

57' 

.0166 

8 

1.0123 

8 

2.0595 

8 

3.1067 

58' 

0169 

9 

1.0297 

9 

2.0769 

9 

3.1241 

59' 

.0172 

60° 

1.0472 

120° 

2.0944 

180° 

3.1416 

60' 

.0175 

Arc  1°  =  0.0174533        Arc  1'  =  0.000290888        Arc  1"  =  0.00000484814 
1  radian  -  57°.295780  =  57°  17'.7468  -  57°  17'  44".806 


MATHEMATICAL  TABLES 
RADIANS  EXPRESSED  IN  DEGREES 


45 


0.01 

0°57 

.64 

36°  67 

1.27 

72°  77 

1  90 

108°.86 

2.53 

144°.% 

Interpolation 

2 

1°.15 

.65 

37°.24 

8 

73°.34 

1 

109°.43 

4 

145°.53 

.0002 

0°.01 

3 

1°.72 

6 

37°.82 

9 

73°.91 

2 

110°.01 

2.55 

146MO 

04 

.02 

4 

2°.29 

7 

38°.39 

1.30 

74°.48 

3 

1KP.58 

6 

146°.68 

06 

.03 

.05 

2°.86 

8 

38°.96 

1 

75°.06 

4 

111°.15 

7 

147°.25 

08 

.05 

6 

3°.44 

9 

39°.53 

2 

75°.63 

1.95 

111°.73 

8 

147°  .82 

.0010 

0°.06 

7 

4°.01 

.70 

40°.1  1 

3 

76°.20 

6 

112°.30 

9 

148°.40 

12 

.07 

8 

4°.58 

1 

40°.68 

4 

76°.78 

7 

112°.87 

2.60 

148°.97 

14 

.08 

9 

5°.  16 

2 

41°.25 

1.35 

77°.35 

8 

113°.45 

149°.54 

16 

.09 

.10 

5°.73 

3 

41°.83 

6 

77°.92 

9 

1I4°.02 

2 

150°.  11 

18 

.10 

1 

6°.30 

4 

42°.40 

7 

78°.50 

2.00 

114°.59 

3 

I50°.69 

.0020 

0°.ll 

2 

6°.88 

.75 

42°.97 

8 

79°.07 

1 

115M6 

4 

151°.26 

22 

.13 

3 

7°.45 

6 

43°.54 

9 

79°.64 

2 

115°.74 

2.65 

15P.83 

24 

.14 

4 

8°.02 

7 

44°.  12 

1.40 

80°.21 

3 

116°.31 

6 

152°.41 

26 

.15 

.15 

8°.59 

8 

44°.69 

1 

80°.79 

4 

116°.88 

7 

152°.98 

28 

.16 

6 

9°.  17 

9 

45°.26 

2 

81°.36 

2.05 

1!7°.46 

8 

153°.55 

.0030 

0°.17 

7 

9°.74 

.80 

45°.84 

3 

81°.93 

6 

118°.03 

9 

154°.  13 

32 

.18 

8 

10°.31 

1 

46°.41 

4 

82°.51 

7 

118°.60 

2.70 

154°.70 

34 

.19 

9 

10°.89 

2 

46°.98 

1.45 

83°.08 

8 

119°.18 

1 

155°.27 

36 

.21 

.20 

1I°.46 

3 

47°.56 

6 

83°.65 

9 

119°.75 

2 

155°.84 

38 

.22 

1 

12°.03 

4 

48°.  13 

7 

84°.22 

2.10 

120°.32 

3 

156°.42 

.0040 

0°.23 

2 

12°.61 

.85 

48°.70 

8 

84°.80 

1 

120°.89 

4 

156°.99 

42 

.24 

3 

13°.18 

6 

49°.27 

9 

85°.37 

2 

121  °.47 

2.75 

157°  .56 

44 

.25 

4 

13°.75 

7 

49°.85 

1.50 

85°.94 

3 

122°.04 

6 

158°.14 

46 

.26 

.25 

14°.32 

8 

50°.42 

1 

86°.52 

4 

122°.61 

7 

158°.7I 

48 

.28 

6 

14°.90 

9 

50°.99 

2 

87°.09 

2.15 

123M9 

8 

159°.28 

.0050 

0°.29 

7 

15°.47 

.90 

5J°.57 

3 

87°.66 

6 

123°.76 

9 

159°.86 

52 

.30 

8 

16°.04 

1 

52°.14 

4 

88°.24 

7 

124°.33 

2.80 

160°.43 

54 

.31 

9 

16°.62 

2 

52°.71 

1.55 

88°.81 

8 

124°.90 

1 

161°.00 

56 

.32 

.30 

17°.19 

3 

53°.29 

6 

89°.38 

9 

125°.48 

2 

16I°.57 

58 

.33 

1 

17°J6 

4 

53°.86 

7 

89°.95 

2.20 

126°.05 

3 

162°.  15 

.0060 

0°.34 

2 

18°.33 

.95 

54°.43 

8 

90°.53 

11 

126°.62 

4 

162  .72 

62 

.36 

3 

18°.91 

6 

55°.00 

9 

91°.10 

127°.20 

2.85 

163°.29 

64 

.37 

4 

19°.48 

7 

55°.58 

1.60 

91°.67 

3 

127°.77 

6 

163°.87 

66 

.38 

.35 

20°.05 

8 

56°.  15 

1 

92°.25 

4 

128°.34 

7 

164°.44 

68 

.39 

6 

20°.63 

9 

56°.72 

2 

92°.82 

2.25 

128°.92 

8 

165°.01 

.0070 

0°.40 

7 

21°.20 

1.00 

57°.30 

3 

93039 

6 

129°.49 

9 

165°.58 

72 

.41 

8 

21°.77 

1 

57°.87 

4 

93°.97 

7 

130°.06 

2.90 

166M6 

74 

.42 

9 

22°.35 

2 

58°.44 

1.65 

94°.54 

8 

130°.6? 

1 

166°.73 

76 

.44 

.40 

22°.92 

3 

59°.01 

6 

95°.  11 

9 

131°.21 

2 

167°.30 

78 

.45 

1 

23°.49 

4 

59°.59 

7 

95°.68 

2.30 

131°.78 

3 

167°.88 

.0080 

0°.46 

2 

24°.06 

1.05 

60°.  16 

8 

96°.26 

1 

132°.35 

4 

168°.45 

82 

.47 

3 

24°.64 

6 

60°.73 

9 

96°.83 

2 

132°.93 

2.95 

169°.02 

84 

.48 

4 

25°.21 

7 

61°.31 

1.70 

97°.40 

3 

133°.50 

6 

169°.60 

86 

.49 

.45 

25°.78 

8 

61°.88 

1 

97°.98 

4 

134°.07 

7 

170°.  17 

88 

.50 

6 

26°.36 

9 

62°.45 

2 

98°.55 

2.35 

134°.65 

8 

170°.74 

.0090 

0°.52 

7 

26°.93 

1.10 

63°.03 

3 

99°.  12 

6 

135°.22 

9 

171°.31 

92 

.53 

8 

27°.50 

1 

63°.60 

4 

99°.69 

7 

135°.79 

3.00 

17P.89 

94 

.54 

9 

28°.07 

2 

64°.  17 

1.75 

100°.27 

8 

136°.36 

172°.46 

96 

.55 

.60 

28°.65 

3 

64°.74 

6 

100°.84 

9 

136°.94 

1 

173°.03 

98 

.56 

| 

29°.22 

4 

65°.32 

7 

10I°.41 

2.40 

137°j51 

3 

173°.61 

2 

29079 

1.15 

65°.89 

8 

10P.99 

1 

138°.08 

4 

174°.  18 

Multiples  of  v 

3 

30°.37 

6 

66°.46 

9 

102°.56 

2 

I38°.66 

3.05 

174°J5 

4 

30°.94 

7 

67°.04 

1.80 

103°.13 

3 

139°.23 

6 

175°.33 

1     3.1416 

180° 

.55 

31°.51 

8 

67°.61 

1 

103°.71 

4 

139°.80 

7 

175°.90 

2    6.2832 

360° 

6 

32°.09 

9 

68°.  18 

2 

104°.28 

2.45 

140°.37 

8 

176°.47 

3    9.4248 

540° 

7 

32°.66 

1.20 

68°.75 

3 

104°.85 

6 

140°.95 

9 

177°.04 

4  12.5664 

720° 

8 

33°.23 

I 

69°.33 

4 

105°.42 

7 

141°.52 

3.10 

177°.62 

5  15.7080 

900° 

9 

33°.80 

2 

69°.90 

1.85 

106°.00 

8 

142°.09 

1 

178°.  19 

6  18.8496 

1080° 

.60 

34°.38 

3 

70°.47 

6 

106°.57 

9 

142°.67 

2 

178°.76 

7  21.9911 

1260° 

1 

34°.95 

4 

71°.05 

7 

107°.  14 

2.50 

143°.24 

3 

179°.34 

8  25.1327 

1440° 

2 

35°.52 

1.25 

71°.62 

8 

107°.72 

1 

143°.81 

V4 

179°.91 

9  28.2743 

1620° 

3 

36°.  10 

6 

72°.  19 

9 

108°.29 

2 

144°.39 

3.15 

180°.48 

10  31.4159 

1800° 

46 


MATHEMATICAL  TABLES 


NATURAL  SINES  AND  COSINES 

Natural  Sines  at  intervals  of  0°.l,  or  6'.     (For  10'  intervals,  see  pp.  52-56) 


M 

o 

o 

0 

o 

B 

o 

0 

o 

. 

• 
P 

=^6o 

(60 

(120  (180 

(240 

(300 

(360 

(420  (480 

(540 

1* 

0.0000 

90° 

0° 

0.0000 

0017 

0035  0052 

0070 

0087 

0105 

0122  0140 

0157 

0175 

89 

17 

1 

0175 

0192 

0209  0227 

0244 

0262 

0279 

0297  0314 

0332 

0349 

88 

17 

2 

0349 

0366 

0384  0401 

0419 

0436 

0454 

0471  0488 

0506 

0523 

87 

17 

3 

0523 

0541 

0558  0576 

0593 

0610 

0628 

0645  0663 

0680 

0698 

86 

17 

4 

0698 

0715 

0732  0750 

0767 

0785 

0802 

0819  0837 

0854 

0.0872 

85 

17 

5 

0.0872 

0889 

0906  0924 

0941 

0958 

0976 

0993  1011 

1028 

1045 

84 

17 

6 

1045 

1063 

1080  1097 

1115 

1132 

1149 

1167  1184 

1201 

1219 

83 

17 

7 

1219 

1236 

1253  1271 

1288 

1305 

1323 

1340  1357 

1374 

1392 

82 

17 

8 

1392 

1409 

1426  1444 

1461 

1478 

1495 

1513  1530 

1547 

1564 

81 

17 

9 

1564 

1582 

1599  1616 

1633 

1650 

1668 

1685  1702 

'1719 

0.1736 

80° 

17 

10° 

0.1736 

1754 

1771  1788 

1805 

1822 

1840 

1857  1874 

1891 

1908 

79 

17 

11 

1908 

1925 

1942  1959 

1977 

1994 

2011 

2028  2045 

2062 

2079 

78 

17 

12 

2079 

2096 

2113  2130 

2147 

2164 

2181 

2198  2215 

2233 

2250 

77 

17 

13 

2250 

2267 

2284  2300 

2317 

2334 

2351 

2368  2385 

2402 

2419 

76 

17 

14 

2419 

2436 

2453  2470 

2487 

2504 

2521 

2538  2554 

2571 

0.2588 

75 

17 

15 

0.2588 

2605 

2622  2639 

2656 

2672 

2689 

2706  2723 

2740 

2756 

74 

17 

16 

2756 

2773 

2790  2807 

2823 

2840 

2857 

2874  2890 

2907 

2924 

73 

17 

17 

2924 

2940 

2957  2974 

2990 

3007 

3024 

3040  3057 

3074 

3090 

72 

17 

18 

3090 

3107 

3123  3140 

3156 

3173 

3190 

3206  3223 

3239 

3256 

71 

17 

19 

3256 

3272 

3289  3305 

3322 

3338 

3355 

3371  3387 

3404 

0.3420 

70° 

16 

20° 

0.3420 

3437 

3453  3469 

3486 

3502 

3518 

3535  3551 

3567 

3584 

69 

16 

21 

3584 

3600 

3616  3633 

3649 

3665 

3681 

3697  3714 

3730 

3746 

68 

16 

22 

3746 

3762 

3778  3795 

3811 

3827 

3843 

3859  3875 

3891 

3907 

67 

16 

23 

3907 

3923 

3939  3955 

3971 

3987 

4003 

4019  4035 

4051 

4067 

66 

16 

24 

4067 

4083 

4099  4115 

4131 

4147 

4163 

4179  4195 

4210 

0.4226 

65 

16 

25 

0.4226 

4242 

4258  4274 

4289 

4305 

4321 

4337  4352 

4368 

4384 

64 

16 

26 

4384 

4399 

4415  4431 

4446 

4462 

4478 

4493  4509 

4524 

4540 

63 

16 

27 

4540 

4555 

4571  4586 

4602 

4617 

4633 

4648  4664 

4679 

4695 

62 

16 

28 

4695 

4710 

4726  4741 

4756 

4772 

4787 

4802  4818 

4833 

4848 

61 

15 

29 

4848 

4863 

4879  4894 

4909 

4924 

4939 

4955  4970 

4985 

0.5000 

60° 

15 

30° 

0.5000 

5015 

5030  5045 

5060 

5075 

5090 

5105  5120 

5135 

5150 

59 

15 

31 

5150 

5165 

5180  5195 

5210 

5225 

5240 

5255  5270 

5284 

5299 

58 

15 

32 

5299 

5314 

5329  5344 

5358 

5373 

5388 

5402  5417 

5432 

5446 

57 

15 

33 

5446 

5461 

5476  5490 

5505 

5519 

5534 

5548  5563 

5577 

5592 

56 

15 

34 

5592 

5606 

5621  5635 

5650 

5664 

5678 

5693  5707 

5721 

0.5736 

55 

14 

35 

0.5736 

5750 

5764  5779 

5793 

5807 

5821 

5835  5850 

5864 

5878 

54 

14 

36 

5878 

5892 

5906  5920 

5934 

5948 

5962 

5976  5990 

6004 

6018 

53 

14 

37 

6018 

6032 

6046  6060 

6074 

6088 

6101 

6115  6129 

6143 

6157 

52 

14 

38 

6157 

6170 

6184  6198 

6211 

6225 

6239 

6252  6266 

6280 

6293 

51 

14 

39 

6293 

6307 

6320  6334 

6347 

6361 

6374 

6388  6401 

6414 

0.6428 

50° 

13 

40° 

0.6428 

6441 

6455  6468 

6481 

6494 

6508 

6521  6534 

6547 

6561 

49 

13 

41 

6561 

6574 

6587  6600 

6613 

6626 

6639 

6652  6665 

6678 

6691 

48 

13 

42 

6691 

6704 

6717  6730 

6743 

6756 

6769 

6782  6794 

6807 

6820 

47 

13 

43 

6820 

6833 

6845  6858 

6871 

6884 

6896 

6909  6921 

6934 

6947 

46 

13 

44 

6947 

6959 

6972  6984 

6997 

7009 

7022 

7034  7046 

7059 

0.7071 

45° 

12 

45° 

0.7071 

=(540 

(480  (420 

(360 

(300 

(240 

(180  (120 

(60 

(00 

i 

(For  graphs,  see  p.  174.) 


Natural  Cosines 


MATHEMATICAL  TABLES 


47 


NATURAL  SINES  AND  COSINES  (continued) 
Natural  Sines  at  intervals  of  0°.l,  or  6'.      (For  10' intervals,  see  pp.  52-56) 


M 

o 

0    °.l 

°2 

03 

°4 

0  5 

°6 

07  03   09 

*d 

a 

=(0/ 

)    (60 

(120 

(180 

(240 

(300 

(360 

(420  (480  (540 

<* 

0.7071 

45° 

45° 

0.7071   7083 

7096 

7108 

7120 

7133 

7145 

7157  7169  7181 

7193 

44 

12 

46 

7193   7206 

7218 

7230 

7242 

7254 

7266 

7278  7290  7302 

7314 

43 

12 

47 

731 

4   7325 

7337 

7349 

7361 

7373 

7385 

7396  7408  7420 

7431 

42 

12 

48 

743 

1   7443 

7455 

7466 

7478 

7490 

7501 

7513  7524  7536 

7547 

41 

12 

49 

7547   7559 

7570 

7581 

7593 

7604 

7615 

7627  7638  7649 

0.7660 

40° 

11 

50° 

0.7660   7672 

7683 

7694 

7705 

7716 

7727 

7738  7749  7760 

7771 

39 

11 

51 

7771   7782 

7793 

7804 

7815 

7826 

7837 

7848  7859  7869 

7880 

38 

11 

52 

788 

0   7891 

7902 

7912 

7923 

7934 

7944 

7955  7965  7976 

7986 

37 

11 

53 

7986   7997 

8007 

8018 

8028 

8039 

8049 

8059  8070  8080 

8090 

36 

10 

54 

8090   8100 

8111 

8121 

8131 

8141 

8151 

8161  8171  8181 

0.8192 

35 

10 

55 

0.8192   8202 

8211 

8221 

8231 

8241 

8251 

8261  8271  8281 

8290 

34 

10 

56 

829 

0   8300 

8310 

8320 

8329 

8339 

8348 

8358  8368  8377 

8387 

33 

10 

57 

8387   8396 

8406 

8415 

8425 

8434 

8443 

8453  8462  8471 

8480 

32 

9 

58 

848 

0   8490 

8499 

8508 

8517 

8526 

8536 

8545  8554  8563 

8572 

31 

9 

59 

8572   8581 

8590 

8599 

8607 

8616 

8625 

8634  8643  8652 

0.8660 

30° 

9 

60° 

0.8660   8669 

8678 

8686 

8695 

8704 

8712 

8721  8729  8738 

8746 

29 

9 

61 

8746   8755 

8763 

8771 

8780 

8788 

8796 

8805  8813  8821 

8829 

28 

8 

62 

882 

9   8838 

8846 

8854 

8862 

8870 

8878 

8886  8894  8902 

8910 

27 

8 

63 

8910   8918 

8926 

8934 

8942 

8949 

8957 

8965  8973  8980 

8988 

26 

8 

64 

8988   8996 

9003 

9011 

9018 

9026 

9033 

9041  9048  9056 

0.9063 

25 

7 

65 

0.9063   9070 

9078 

9085 

9092 

9100 

9107 

9114  9121  9128 

9135 

24 

7 

66 

913 

5   9143 

9150 

9157 

9164 

9171 

9178 

9184  9191  9198 

9205 

23 

7 

67 

9205   9212 

9219 

9225 

9232 

9239 

9245 

9252  9259  9265 

9272 

22 

7 

68 

927 

2   9278 

9285 

9291 

9298 

9304 

9311 

9317  9323  9330 

9336 

21 

6 

69 

9336   9342 

9348 

9354 

9361 

9367 

9373 

9379  9385  9391 

0.9397 

20° 

6 

70° 

0.9397   9403 

9409 

9415 

9421 

9426 

9432 

9438  9444  9449 

9455 

19 

6 

71 

945 

5   9461 

9466 

9472 

9478 

9483 

9489 

9494  9500  9505 

9511 

18 

6 

72 

951 

1   9516 

9521 

9527 

9532 

9537 

9542 

9548  9553  9558 

9563 

17 

5 

73 

.  9563   9568 

9573 

9578 

9583 

9588 

9593 

9598  9603  9608 

9613 

16 

5 

74 

9613   9617 

9622 

9627 

9632 

9636 

9641 

9646  9650  9655 

0.9659 

15 

5 

75 

0.9659   9664 

9668 

9673 

9677 

9681 

9686 

9690  9694  9699 

9703 

14 

4 

76 

970 

3   9707 

9711 

9715 

9720 

9724 

9728 

9732  9736  9740 

9744 

13 

4 

77 

9744   9748 

9751 

9755 

9759 

9763 

9767 

9770  9774  9778 

9781 

12 

4 

78 

978 

1   9785 

9789 

9792 

97% 

9799 

9803 

9806  9810  9813 

9816 

11 

3 

79 

9816   9820 

9823 

9826 

9829 

9833 

9836 

9839  9842  9845 

0.9848 

10° 

3 

80° 

0.9848   9851 

9854 

9857 

9860 

9863 

9866 

9869  9871  9874 

9877 

9 

3 

81 

9877   9880 

9882 

9885 

9888 

9890 

9893 

9895  9898  9900 

9903 

8 

3 

82 

990 

3   9905 

9907 

9910 

9912 

9914 

9917 

9919  9921  9923 

9925 

7 

2 

83 

992 

5   9928 

9930 

9932 

9934 

9936 

9938 

9940  9942  9943 

9945 

6 

2 

84 

9945   9947 

9949 

9951 

9952 

9954 

9956 

9957  9959  9960 

0.9962 

5 

2 

85 

0.9962   9963 

9965 

9966 

9968 

9969 

9971 

9972  9973  9974 

9976 

4 

1 

86 

997 

6   9977 

9978 

9979 

9980 

9981 

9982 

9983  9984  9985 

9986 

3 

1 

87 

9986   9987 

9988 

9989 

9990 

9990 

9991 

9992  9993  9993 

9994 

2 

1 

88 

999 

4   9995 

9995 

9996 

9996 

9997 

9997 

9997  9998  9998 

0.9998 

1 

0 

89 

0.9998   9999 

9999 

9999 

9999 

0000 

0000 

0000  0000  0000 

1.  0000 

0° 

0 

90° 

1.0000 

=  (540 

(480 

(420 

(360 

(300 

(240 

(180  (120  (60 

(00 

$ 

Q 

Natural  Cosines 


48 


MATHEMATICAL  TABLES 


NATURAL  TANGENTS  AND  COTANGENTS 

Natural  Tangents  at  intervals  of  0°.l,  or  6'.    (For  10'  intervals,  see  pp.  52-56) 


ff 

°.0 

M 

°2 

°3   °4 

05 

0(j 

07   03 

°9 

*  • 

Q 

-coo 

(60 

(12') 

(18')  (24') 

(30') 

(360 

(420  (48') 

(54') 

IS 

0.0000 

90° 

0° 

0.0000 

0017 

0035 

0052  0070 

0087 

0105 

0122  0140 

0157 

0175 

89 

17 

1 

0175 

0192 

0209 

0227  0244 

0262 

0279 

0297  0314 

0332 

0349 

88 

17 

2 

0349 

0367 

0384 

0402  0419 

0437 

0454 

0472  0489 

0507 

0524 

87 

17 

3 

0524 

0542 

0559 

0577  0594 

0612 

0629 

0647  0664 

0682 

0699 

86 

18 

4 

0699 

0717 

0734 

0752  0769 

0787 

0805 

0822  0840 

0857 

0.0875 

85 

18 

5 

0.0875 

0892 

0910 

0928  0945 

0963 

0981 

0998  1016 

1033 

1051 

84 

18 

6 

1051 

1069 

1086 

1104  1122 

1139 

1157 

1175  1192 

1210 

1228 

83 

18 

7 

1228 

1246 

1263 

1281  1299 

1317 

1334 

1352  1370 

1388 

1405 

82 

18 

8 

1405 

1423 

1441 

1459  1477 

1495 

1512 

1530  1548 

1566 

1584 

81 

18 

9 

1584 

1602 

1620 

1638  1655 

1673 

1691 

1709  1727 

1745 

0.1763 

80° 

18 

10° 

0.1763 

1781 

1799 

1817  1835 

1853 

1871 

1890  1908 

1926 

1944 

79 

18 

It 

1944 

1962 

1980 

1998  2016 

2035 

2053 

2071  2089 

2107 

2126 

78 

18 

12 

2126 

2144 

2162 

2180  2199 

2217 

2235 

2254  2272 

2290 

2309 

77 

18 

13 

2309 

2327 

2345 

2364  2382 

2401 

2419 

2438  2456 

2475 

2493 

76 

18 

14 

2493 

2512 

2530 

2549  2568 

2586 

2605 

2623  2642 

2661 

0.2679 

75 

19 

15 

0.2679 

2698 

2717 

2736  2754 

2773 

2792 

2811  2830 

2849 

2867 

74 

19 

16 

2867 

2886 

2905 

2924  2943 

2962 

2981 

3000  3019 

3038 

3057 

73 

19 

17 

3057 

3076 

3096 

3115  3134 

3153 

3172 

3191  3211 

3230 

3249 

72 

19 

18 

3249 

3269 

3288 

3307  3327 

3346 

3365 

3385  3404 

3424 

3443 

71 

19 

19 

3443 

3463 

3482 

3502  3522 

3541 

3561 

3581  3600 

3620 

0.3640 

70° 

20 

20° 

0.3640 

3659 

3679 

3699  3719 

3739 

3759 

3779  3799 

3819 

3839 

69 

20 

21 

3839 

3859 

3879 

3899  3919 

3939 

3959 

3979  4000 

4020 

4040 

68 

20 

22 

4040 

4061 

4081 

4101  4122 

4142 

4163 

4183  4204 

4224 

4245 

67 

21 

23 

4245 

4265 

4286 

4307  4327 

4348 

4369 

4390  4411 

4431 

4452 

66 

21 

24 

4452 

4473 

4494 

4515  4536 

4557 

4578 

4599  4621 

4642 

0.4663 

65 

21 

25 

0.4663 

4684 

4706 

4727  4748 

4770 

4791 

4813  4834 

4856 

4877 

64 

21 

26 

4877 

4899 

4921 

4942  4964 

4986 

5008 

5029  5051 

5073 

5095 

63 

22 

27 

5095 

5117 

5139 

5161  5184 

5206 

5228 

5250  5272 

5295 

5317 

62 

22 

28 

5317 

5340 

5362 

5384  5407 

5430 

5452 

5475  5498 

5520 

5543 

61. 

23 

29 

5543 

5566 

5589 

5612  5635 

5658 

5681 

5704  5727 

5750 

0.5774 

60° 

23 

30° 

0.5774 

5797 

5820 

5844  5867 

5890 

5914 

5938  5961 

5985 

6009 

59 

24 

31 

6009 

6032 

6056 

6080  6104 

6128 

6152 

6170  6200 

6224 

6249 

58 

24 

32 

6249 

6273 

6297 

6322  6346 

6371 

6395 

6420  6445 

6469 

6494 

57 

25 

33 

6494 

6519 

6544 

6569  6594 

6619 

6644 

6669  6694 

6720 

6745 

56 

25 

34 

6745 

6771 

6796 

6822  6847 

6873 

6899 

6924  6950 

6976 

0.7002 

55 

26 

35 

0.7002 

7028 

7054 

7080  7107 

7133 

7159 

7186  7212 

7239 

7265 

54 

26 

36 

7265 

7292 

7319 

7346  7373 

7400 

7427 

7454  7481 

7508 

7536 

53 

27 

37 

7536 

7563 

7590 

7618  7646 

7673 

7701 

7729  7757 

7785 

7813 

52 

28 

38 

7813 

7841 

7869 

7898  7926 

7954 

7983 

8012  8040 

8069 

8098 

51 

28 

39 

8098 

8127 

8156 

8185  8214 

8243 

8273 

8302  8332 

8361 

0.8391 

50° 

29 

40° 

0.8391 

8421 

8451 

8481  8511 

8541 

8571 

8601  8632 

8662 

8693 

49 

30 

41 

8693 

8724 

8754 

8785  8816 

8847 

8878 

8910  8941 

8972 

9004 

48 

31 

42 

9004 

9036 

9067 

9099  9131 

9163 

9195 

9228  9260 

9293 

9325 

47 

32 

43 

9325 

9358 

9391 

9424  9457 

9490 

9523 

9556  9590 

9623 

0.9657 

46 

33 

44 

0.9657 

9691 

9725 

9759  9793 

9827 

9861 

9896  9930 

9965 

1.0000 

45° 

34 

45° 

1.0000 

*(549') 

(480 

(420  (36') 

(300 

(240 

(18')  (120 

(60 

(00 

1 

(For  graphs,  see  p.  174.) 


Natural  Cotangents 


MATHEMATICAL  TABLES 


49 


NATURAL  TANGENTS  AND  COTANGENTS  (continued) 

Natural  Tangents  at  intervals  of  0°.  1,  or  6'.    (For  10'  intervals,  see  pp.  52-56) 


i 

a=Y(V 

)        (60 

(120 

(180 

(240 

(300 

(360    (420 

(480 

(540 

diff! 

1.0000 

45° 

45° 

1.0000       0035 

0070 

0105 

0141 

0176 

0212    0247 

0283 

0319    0355 

44 

35 

46 

035 

5       0392 

0428 

0464 

0501 

0538 

0575    0612 

0649 

0686    0724 

43 

37 

47 

072 

4        0761 

0799 

0837 

0875 

0913 

0951     0990 

1028 

1067    1106 

42 

38 

48 

1106        1145 

1184 

1224 

1263 

1303 

1343     1383 

1423 

1463    1504 

41 

40 

49 

1504        1544 

1585 

1626 

1667 

1708 

1750    1792 

1833 

18751.1918 

40° 

41 

50° 

1.1918        1960 

2002 

2045 

2088 

2131 

2174    2218 

2261 

2305    2349 

39 

43 

51 

234 

9        2393 

2437 

2482 

2527 

2572 

2617    2662 

2708 

2753    2799 

38 

45 

52 

2799        2846 

2892 

2938 

2985 

3032 

3079    3127 

3175 

3222    3270 

37 

47 

53 

327 

0        3319 

3367 

3416 

3465 

3514 

3564    3613 

3663 

3713    3764 

36 

49 

54 

3764        3814 

3865 

3916 

3968 

4019 

4071    4124 

4176 

42291.4281 

35 

52 

55 

1  .4281        4335 

4388 

4442 

44% 

4550 

4605    4659 

4715 

4770    4826 

34 

55 

56 

4826        4882 

4938 

4994 

5051 

5108 

5166    5224 

5282 

5340    5399 

33 

57 

57 

539 

9        5458 

5517 

5577 

5637 

5697 

5757    5818 

5880 

5941     6003 

32 

60 

58 

6003        6066 

6128 

6191 

6255 

6319 

6383    6447 

6512 

6577    6643 

31 

64 

59 

1.6643        6709 

6775 

6842 

6909 

6977 

7045    7113 

7182 

7251  1.7321 

30° 

67 

60° 

1.732       1.739 

1.746 

1.753 

1.760 

1.76} 

1.775  1.782 

1.789 

1.797    1.804 

29 

7 

61 

1.80 

4       1.811 

1.819 

1.827 

1.834 

1.842 

1.849  1.857 

1.865 

1.873    1.881 

28 

8 

62 

1.88 

1        1.889 

1.897 

1.905 

1.913 

1.921 

1.929  1.937 

1.946 

1.954   1.963 

27 

8 

63 

1.963       1.971 

1.980 

1.988 

1.997 

2.006 

2.014  2023 

2.032 

2.041    2.050 

26 

9 

64 

2.050      2.059 

2.069 

2.078 

2.087 

2.097 

2.1062.116 

2.125 

2.135  2.145 

25 

9 

65 

2.145       2.154 

2.164 

2.174 

2.184 

2.194 

2.204  2.215 

2.225 

2.236  2.246 

24 

10 

66 

2.24 

6       2.257 

2.267 

2.278 

2.289 

2.300 

2.311  2.322 

2.333 

2.344  2.356 

23 

11 

67 

2.356       2.367 

2.379 

2.391 

2.402 

2.414 

2.426  2.438 

2.450 

2.463   2.475 

22 

12 

68 

2.47 

5       2.488 

2.500 

2.513 

2.526 

2.539 

2.552  2.565 

2.578 

2.592  2.605 

21 

13 

69 

2.605       2.619 

2.633 

2.646 

2.660 

2.675 

2.689  2.703 

2.718 

2.733   2.747 

20° 

14 

70° 

2.747       2.762 

2.778 

2.793 

2.808 

2.824 

2.840  2.856 

2.872 

2.888   2.904 

19 

16 

71 

2.904       2.921 

2.937 

2.954 

2.971 

2.989 

3.006  3.024 

3.042 

3.060  3.078 

18 

17 

72 

3.07 

8       3.096 

3.115 

3.133 

3.152 

3.172 

3.191  3.211 

3.230 

3.251    3.271 

17 

19 

73 

3.271       3.291 

3.312 

3.333 

3.354 

3.376 

3.398  3.420 

3.442 

3.465   3.487 

16 

22 

74 

3.487       3.511 

3.534 

3.558 

3.582 

3.606 

3.630  3.655 

3.681 

3.706   3.732* 

15 

24 

75 

3.732       3.758 

3.785 

3.812 

3.839 

3.867 

3.895  3.923 

3.952 

3.981    4.011 

14 

28 

76 

4.01 

1       4.041 

4.071 

4.102 

4.134 

4.165 

4.198  4.230 

4.264 

4.297  4.331 

13 

32 

77 

4.331       4.366 

4.402 

4.437 

4.474 

4.511 

4.548  4.586 

4.625 

4.665   4.705 

12 

37 

78 

4.70 

5       4.745 

4.787 

4.829 

4.872 

4.915 

4.959  5.005 

5.050 

5.097   5.145 

11 

44 

79 

5.145       5.193 

5.242 

5.292 

5.343 

5.396 

5.449  5.503 

5.558 

5.614  5:671 

10° 

53 

80° 

5,671       5.730 

5.789 

5.850 

5.912 

5.976 

6.041  6.107 

6.174 

6.243   6.314 

9 

81 

6.314       6.386 

6.460 

6.535 

6.612 

6.691 

6.772  6  855 

6.940 

7.026   7.115 

8 

82 

7.11 

5       7.207 

7.300 

7.396 

7.495 

7.596 

7.700  7.806 

7.916 

8.028  8.144 

7 

83 

8.14 

4       8.264 

8.386 

8.513 

8.643 

8.777 

8.915  9.058 

9.205 

9.357   9.514 

6 

84 

9.514       9.677 

9.845 

10.02 

10.20 

10.39 

10.58  10.78 

10.99 

11.20   11.43 

5 

85 

11.43       11.66 

11.91 

12.16 

12.43 

12.71 

13.00  1330 

13.62 

13.95    14.30 

4 

86 

14.3 

[)       14.67 

15.06 

15.46 

15.89 

16.35 

16.83  17.34 

17.89 

18.46    19.08 

3 

87 

19.a 

8       19.74 

20.45 

21.20 

22.02 

22.90 

23.86  24^90 

26.03 

27.27   28.64 

2 

88 

28.64       30.14 

31.82 

33.69 

35.80 

38.19 

40.92  44.07 

47.74 

52.08   57.29 

1 

89 

57.29       63.66 

71.62 

81.85 

95.49 

114.6 

143.2  191.0 

286.5 

573.0     oo 

0° 

90° 

00 

=(540 

(480 

(420 

(360 

(300 

(240     (180 

(120 

(60     (0*0 

i 

Natural  Cotangents 


50 


MATHEMATICAL  TABLES 


NATURAL  SECANTS  AND  COSECANTS 

Natural  Secants  at  intervals  of  0°.  1,  or  6'.     (For  10'  intervals,  see  pp.  52-56) 


i 

-(V 

(60* 

(120 

(180  (240 

(3V) 

(360 

(420 

(480  (540 

Avg. 
diff. 

1.0000 

90° 

0° 

1.0001 

3   0000 

0000 

0000  0000 

0000 

0001 

0001 

0001  0001 

0002 

89 

0 

1 

000 

I   0002 

0002 

0003  0003 

0003 

0004 

0004 

0005  0006 

0006 

88 

0 

2 

000 

i   0007 

0007 

0008  0009 

0010 

0010 

0011 

0012  0013 

0014 

87 

3 

001' 

\   0015 

0016 

0017  0018 

0019 

0020 

0021 

0022  0023 

0024 

86 

1 

4 

002' 

I   0026 

0027 

0028  0030 

0031 

0032 

0034 

0035  0037 

1.0038 

85 

1 

5 

1.003 

J   0040 

0041 

0043  0045 

0046 

0048 

0050 

0051  0053 

0055 

84 

2 

6 

005. 

>   0057 

0059 

0061  0063 

0065 

0067 

0069 

0071  0073 

0075 

83 

2 

7 

007. 

>   0077 

0079 

0082  0084 

0086 

0089 

0091 

0093  0096 

0098 

82 

2 

8 

009* 

)   0101 

0103 

0106  0108 

0111 

0114 

0116 

0119  0122 

0125 

81 

3 

9 

01  2f 

>   0127 

0130 

0133  0136 

0139 

0142 

0145 

0148  0151 

1.0154 

80° 

3 

10° 

\   0157 

0161 

0164  0167 

0170 

0174 

0177 

0180  0184 

0187 

79 

3 

11 

018} 

'   0191 

0194 

0198  0201 

0205 

0209 

0212 

0216  0220 

0223 

78 

4 

12 

022: 

I   0227 

0231 

0235  0239 

0243 

0247 

0251 

0255  0259 

0263 

77 

4 

13 

026: 

I   0267 

0271 

0276  0280 

0284 

0288 

0293 

0297  0302 

0306 

76 

4 

14 

030* 

>   0311 

0315 

0320  0324 

0329 

0334 

0338 

0343  0348 

1.0353 

75 

5 

15 

1  .035: 

I   0358 

0363 

0367  0372 

0377 

*0382 

0388 

0393  0398 

0403 

74 

5 

16 

040: 

5   0408 

0413 

0419  0424 

0429 

0435 

0440 

0446  0451 

0457 

73 

5 

17 

0453 

'   0463 

0468 

0474  0480 

0485 

0491 

0497 

0503  0509 

0515 

72 

6 

18 

051f 

0521 

0527 

0533  0539 

0545 

0551 

0557 

0564  0570 

0576 

71 

6 

19 

057* 

0583 

0589 

0595  0602 

0608 

0615 

0622 

0628  0635 

1.0642 

70° 

7 

20° 

1.064; 

'   0649 

0655 

0662  0669 

0676 

0683 

0690 

0697  0704 

0711 

69 

7 

21 

071 

0719 

0726 

0733  0740 

0748 

0755 

0763 

0770  0778 

0785 

68 

7 

22 

078f 

0793 

0801 

0808  0816 

0824 

0832 

0840 

0848  0856 

0864 

67 

8 

23 

0864 

0872 

0880 

0888  0896 

0904 

0913 

0921 

0929  0938 

0946 

66 

8 

24 

094* 

0955 

0963 

0972  0981 

0989 

0998 

1007 

1016  1025 

1.1034 

65 

9 

25 

1.103' 

f   1043 

1052 

1061  1070 

1079 

1089 

1098 

1107  1117 

1126 

64 

9 

26 

112* 

>   1136 

1145 

1155  1164 

1174 

1184 

1194 

1203  1213 

1223 

63 

10 

27 

1222 

1233 

1243 

1253  1264 

1274 

1284 

1294 

1305  1315 

1326 

62 

10 

28 

132* 

1336 

1347 

1357  1368 

1379 

1390 

1401 

1412  1423 

1434 

61 

11 

29 

1434 

1445 

1456 

1467  1478 

1490 

1501 

1512 

1524  1535 

1.1547 

60° 

11 

30° 

1.1543 

'   1559 

1570 

1582  1594 

1606 

1618 

1630 

1642  1654 

1666 

59 

12 

31 

166* 

1679 

1691 

1703  1716 

1728 

1741 

1753 

1766  1779 

1792 

58 

13 

32 

179; 

1805 

1818 

1831  1844 

.  1857 

1870 

1883 

1897  1910 

1924 

57 

13 

33 

1924 

1937 

1951 

1964  1978 

1992 

2006 

2020 

2034  2048 

2062 

56 

14 

34 

206; 

2076 

2091 

2105  2120 

2134 

2149 

2163 

2178  2193 

1.2208 

55 

15 

35 

1.220* 

2223 

2238 

2253  2268 

2283 

2299 

2314 

2329  2345 

2361 

54 

15 

36 

2361 

2376 

2392 

2408  2424 

2440 

245* 

2472 

2489  2505 

2521 

53 

16 

37 

2521 

2538 

2554 

2571  2588 

2605 

2622 

2639 

2656  2673 

2690 

52 

17 

38 

269C 

2708 

2725 

2742  2760 

2778 

2796 

2813 

2831  2849 

2868 

51 

18 

39 

2866 

2886 

2904 

2923  2941 

2960 

2978 

2997 

3016  3035 

1.3054 

60° 

19 

40° 

1.3054 

3073 

3093 

3112  3131 

3151 

3171 

3190 

3210  3230 

3250 

49 

20 

41 

325C 

3270 

3291 

3311  3331 

3352 

3373 

3393 

34  M  3435 

3456 

48 

21 

42 

345* 

3478 

3499 

3520  3542 

3563 

3585 

3607 

3629  3651 

3673 

47 

22 

43 

3673 

3696 

3718 

3741  3763 

3786 

3809 

3832 

3855  3878 

3902 

46 

23 

44 

3902 

3925 

3949 

3972  3996 

4020 

4044 

4069 

4093  4118 

1.4142 

45° 

24 

45° 

1.4142 

t 

, 

=  (540 

(480 

(420  (360 

(300 

(240 

(180 

(120  (60 

(00 

Q 

(For  graphs,  see  p.  174.) 


Natural  Cosecants 


MATHEMATICAL  TABLES 


51 


NATURAL  SECANTS  AND  COSECANTS   (continued) 

Natural  Secants  at  intervals  of  0°.l,  or  6'.     (For  10'  intervals,  see  pp.  52-56) 


« 

°.o 

°.l 

°.a 

°.3 

°.4 

°.5 

°.6       °.7 

°.8     °.9 

Avg. 

Q 

=(00 

(60 

(120 

(180 

(240 

(300 

(360    (420 

(480  (540 

diff. 

1.4142 

45° 

45° 

1.4142 

4167 

4192 

4217 

4242 

4267 

4293    4318 

4344    4370 

4396 

44 

25 

46 

4396 

4422 

4448 

4474 

4501 

4527 

4554    4581 

4608    4635 

4663 

43 

27 

47 

4663 

4690 

4718 

4746 

4774 

4802 

4830    4859 

4887    4916 

4945 

42 

28 

48 

4945 

4974 

5003 

5032 

5062 

5092 

5121    5151 

5182    5212 

5243 

41 

30 

49 

5243 

5273 

5304 

5335 

5366 

5398 

5429    5461 

5493    5525 

1.5557 

40° 

31 

60° 

1.5557 

5590 

5622 

5655 

5688 

5721 

5755    5788 

5822    5856 

5890 

39 

33 

51 

5890 

5925 

5959 

5994 

6029 

6064 

6099    6135 

6171     6207 

6243 

38 

35 

52 

6243 

6279 

6316 

6353 

6390 

6427 

6464    6502 

6540    6578 

6616 

37 

37 

53 

6616 

6655 

6694 

6733 

6772 

6812 

6852    6892 

6932    6972 

7013 

36 

40 

54 

7013 

7054 

7095 

7137 

7179 

7221 

7263    7305 

7348    7391 

1.7434 

35 

42 

55 

1.7434 

7478 

7522 

7566 

7610 

7655 

7700    7745 

7791     7837 

7883 

34 

45 

56 

7883 

7929 

7976 

8023 

8070 

8118 

8166    8214 

8263    8312 

8361 

33 

48 

57 

8361 

8410 

8460 

8510 

8561 

8612 

8663    8714 

8766    8818 

8871 

32 

51 

58 

8871 

8924 

8977 

9031 

9084 

9139 

9194    9249 

9304    9360 

1.9416 

31 

54 

59 

1.9416 

9473 

9530 

9587 

9645 

9703 

9762    9821 

9880    9940 

2.0000 

30° 

58 

60° 

2.00C 

2.006 

2.012 

2.018 

2.025 

2.031 

2.037  2.043 

2.050  2.056 

2.063 

29 

6 

61 

2.063 

2.069 

2.076 

2.082 

2.089 

2.096 

2.103  2.109 

2.116  2.123 

2.130 

28 

7 

62 

2.13C 

2.137 

2.144 

2.151 

2.158 

2.166 

2.173  2.180 

2.188  2.195 

2.203 

27 

7 

63 

2.203 

2.210 

2.218 

2.226 

2.233 

2.241 

2.249  2.257 

2.265  2.273 

2281 

26 

8 

64 

2.281 

2.289 

2.298 

2.306 

2.314 

2.323 

2.331   2.340 

2349  2357 

2366 

25 

8 

65 

2.36< 

2375 

2.384 

2.393 

2.402 

2.411 

2.421   2.430 

2.439  2.449 

2.459 

24 

9 

66 

2.45< 

2.468 

2.478 

2.488 

2.498 

2.508 

2.518  2.528 

2.538  2.549 

2.559 

23 

10 

67 

2.55< 

2.570 

2.581 

2.591 

2.602 

2.613 

2624  2.635 

2.647  2.658 

2.669 

22 

11 

68 

2.66' 

2.681 

2.693 

2.705 

2.716 

2.729 

2.741   2.753 

2.765  2.778 

2.790 

21 

12 

69 

2.79( 

2.803 

2.816 

2.829 

2.842 

2.855 

2.869  2.882 

2.896  2.910 

2.924 

20° 

13 

70° 

2.92^ 

\       2.938 

2.952 

2.967 

2.981 

2.996 

3.011   3.026 

3.041   3.056 

3.072 

19 

15 

71 

3.07; 

3.087 

3.103 

3.119 

3.135 

3.152 

3.168  3.185 

3.202  3.219 

3.236 

18 

16 

72 

3.23( 

»       3.254 

3.271 

3.289 

3.307 

3326 

3.344  3.363 

3.382  3.401 

3.420 

17 

18 

73 

3.42C 

3.440 

3.460 

3.480 

3.500 

3.521 

3.542  3.563 

3.584  3.606 

3.628 

16 

21 

74 

3.62* 

I      3.650 

3.673 

3.695 

3.719 

3.742 

3.766  3.790 

3.814  3.839 

3.864 

15 

24 

75 

3.86^ 

I       3.889 

3.915 

3.941 

3.967 

3.994 

4.021   4.049 

4.077  4.105 

4.134 

14 

27 

76 

4.13^ 

4.163 

4.192 

4.222 

4.253 

4.284 

4.315  4.347 

4379  4.412 

4.445 

13 

31 

77 

4.44! 

4.479 

4.514 

4.549 

4.584 

4.620 

4.657  4.694 

4.732  4.771 

4.810 

12 

36 

78 

4.8H 

1       4.850 

4.890 

4.931 

4.973 

5.016 

5.059  5.103 

5.148  5.194 

5.241 

11 

43 

79 

5.24 

5.288 

5337 

5.386 

5.436 

5.487 

5.540  5.593 

5.647  5.702 

5.759 

10° 

52 

80° 

5.75C 

>       51816 

5.875 

5.935 

5.996 

6.059 

6.123  6.188 

6.255  6.323 

6392 

9 

81 

6.39; 

!       6.464 

6.537 

6.611 

6.687 

6.765 

6.845  6.927 

7.011   7.097 

7.185 

8 

82 

7.18f 

•       7.276 

7.368 

7.463 

7.561 

7.661 

7.764  7.870 

7.979  8.091 

8.206 

7 

83 

8.2W 

»       8.324 

8.446 

8.571 

8.700 

8.834 

8.971   9.113 

9.259  9.411 

9.567 

6 

84 

9.563 

9.728 

9.895 

10.07 

1025 

10.43 

10.63  10.83 

11.03  11.25 

11.47 

5 

85 

11.42 

'       11.71 

11.95 

12.20 

12.47 

12.75 

13.03  13.34 

13.65  13.99 

1434 

4 

86 

14.3' 

\       14.70 

15.09 

15.50 

15.93' 

16.38 

16.86  17.37 

17.91    18.49 

19.11 

3 

87 

19.1 

19.77 

20.47 

21.23 

22,04 

22.93 

23.88  24.92 

26.05  27.29 

28.65 

2 

88 

28.6! 

»       30.16 

31.84 

33.71 

35.81 

38.20 

40.93  44.08 

47.75  52.09 

57.30 

89 

573( 

)       63.66 

71.62 

81.85 

95.49 

114.6 

143.2  191.0 

286.5  573.0 

00 

0° 

90° 

oo 

°.9 

°.8 

°.7 

°.6 

°.5 

M 

=(540 

(480 

(420 

(360 

(300 

(240     (180 

(120     (60 

(oo 

a 

Natural  Cosecants 


52 


MATHEMATICAL  TABLES 


TRIGONOMETRIC  FUNCTIONS  (at  intervals  of  10') 

Annex  -10  in  columns  marked  *.     (For  O.°l  intervals,  see  pp.  46-51) 


De- 
grees 

Ra- 
dians 

Sines 

Cosines 

Tangents 

Cotangents 

Nat.  Log.* 

Nat.  Log.* 

Nat.  Log.* 

Nat.  Log. 

0°  (XX 

0.0000 

.0000   « 

t.OOOO  0.0000 

.0000 

CO        00 

1.5708 

90°  00* 

10 

00029 

.0029  7.4637 

1.0000  .0000 

.0029  7.4637 

343.77  2.5363 

1.5679 

50 

20 

00058 

.0058   .7648 

1.0000  .0000 

.0058   .7648 

171.89  .2352 

1.5650 

40 

30 

00087 

.0087   .9408 

1.0000  .0000 

.0087   .9409 

114.59  .0591 

1.5621 

30 

40 

0.0116 

.0116  8.0658 

0.9999  .0000 

.0116  8.0658 

85.940  1.9342 

1.5592 

20 

50 

0.0145 

.0145   .1627 

.9999  .0000 

.0145   .1627 

68.750  .8373 

1.5563 

10 

1°  00' 

0.0175 

.0175  8.2419 

.9998  9.9999 

.0175  8.2419 

57.290  1.7581 

1.5533 

89°  00' 

10 

0.0204 

.0204   .3088 

.9998  .9999 

.0204   .3089 

49.104  .6911 

1.5504 

50 

20 

0.0233 

.0233   .3668 

.9997  .9999 

.0233   .3669 

42.964  .6331 

1.5475 

40 

30 

0.0262 

.0262   .4179 

.9997  .9999 

.0262   .4181 

38.188  .5819 

1.5446 

30 

40 

0.0291 

.0291   .4637 

.9996  .9998 

.0291   .4638 

34.368  .5362 

1.5417 

20 

50 

0.0320 

.0320   .5050 

.9995  .9998 

.0320   .5053 

31.242  .4947 

1.5388 

10 

2°  (XX 

0.0349 

.0349  8.5428 

.9994  9.9997 

.0349  8.5431 

28.636  1.4569 

1.5359 

88°  00' 

10 

0.0378 

.0378   .5776 

.9993  .9997 

.0378   .5779 

26.432  .4221 

1.5330 

50 

20 

0.0407 

.0407   .6097 

.9992  .9996 

.0407   .6101 

24.542  .3899 

1.5301 

40 

30 

0.0436 

.0436   .6397 

.9990  .9996 

.0437   .6401 

22.904  .3599 

1.5272 

30 

40 

0.0465 

.0465   .6677 

.9989  .9995 

.0466   .6682 

21.470  .3318 

1  .5243 

20 

50 

0.0495 

.0494   .6940 

.9988  .9995 

.0495   .6945 

20.206  .3055 

1.5213 

10 

3°  (XX 

0.0524 

.0523  8.7188 

.9986  9.9994 

.0524  8.7194 

19.081  1.2806 

1.5184 

87°  00' 

10 

0.0553 

.0552   .7423 

.9985  .9993 

.0553   .7429 

18.075  .2571 

1.5155 

50 

20 

0.0582 

.0581   .7645 

.9983  .9993 

.0582   .7652 

17.169  .2348 

1.5126 

40 

30 

0.061  1 

.0610   .7857 

.9981  .9992 

.0612   .7865 

16.350  .2135 

1.5097 

30 

40 

0.0640 

.0640   .8059 

.9980  .9991 

.0641   .8067 

15.605  .1933 

1.5068 

20 

50 

0.0669 

.0669   .8251 

.9978  .9990 

.0670  .8261 

14.924  .1739 

1.5039 

10 

4°  (XX 

0.0698 

.0698  8.8436 

.9976  9.9989 

.0699  8.8446 

14.301  1.1554 

1.5010 

86°  00 

10 

0.0727 

.0727   .8613 

.9974  .9989 

.0729   .8624 

13.727  .1376 

1.4981 

50 

20 

0.0756 

.0756   .8783 

.9971  .9988 

.0758   .8795 

13.197  .1205 

1.4952 

40 

30 

0.0785 

.0785   .8946 

.9969  .9987 

.0787   .8960 

12.706  .1040 

1.4923 

30 

40 

0.0814 

.0814   .9104 

.9967  .9986 

.0816   .9118 

12.251  .0882 

1.4893 

20 

50 

0.0844 

.0843   .9256 

,9964  .9985 

.0846   .9272 

11.826  .0728 

1.4864 

10 

5°  (XX 

0.0873 

.0872  8.9403 

.9962  9.9983 

.0875  8.9420 

11.430  1.0580 

1.4835 

85°  00' 

10 

0.0902 

.0901   .9545 

.9959  .9982 

.0904   .9563 

11.059  .0437 

1.4806 

50 

20 

0.0931 

.0929   .9682 

.9957  .9981 

.0934   .9701 

10.712  .0299 

1.4777 

40 

30 

0.0960 

.0958   .9816 

.9954  .9980 

.0963   .9836 

10.385  .0164 

1.4748 

30 

40 

0.0989 

.0987   .9945 

.9951  .9979 

.0992   .9966 

10.078  .0034 

1.4719 

'  20 

50 

0.1018 

.1016  9.0070 

.9948  .9977 

.1022  9.0093 

9.7882  0.9907 

1.4690 

10 

6°  00' 

0.1047 

.1045  9.0192 

.9945  9.9976 

.1051  9.0216 

9.5144  0.9784 

1.4661 

84°  00' 

10 

0.1076 

.1074   .0311 

.9942  .9975 

.1080   .0336 

9.2553  .9664 

1.4632 

50 

20 

0.1105 

.1103   .0426 

.9939  .9973 

.1110   .0453 

9.0098  .9547 

1.4603 

40 

30 

0.1134 

.1132   .0539 

.9936  .9972 

.1139   .0567 

8.7769  .9433 

1.4574 

30 

40 

0.1164 

.1161   .0648 

.9932  .9971 

.1169   .0678 

8.5555  .9322 

1.4544 

20 

50 

0.1193 

.1190   .0755 

.9929  .9969 

.1198   .0786 

8.3450  .9214 

1.4515 

10 

7°  00' 

0.1222 

.1219  9.0859 

.9925  9.9968 

.1228  9.0891 

8.1443  0.9109 

1.4486 

83°  00' 

10 

0.1251 

.1248   .0961 

.9922  .9966 

.1257   .0995 

7.9530  .9005 

1.4457 

50 

20 

0.1280 

.1276   .1060 

.9918  .9964 

.1287   .1096 

7.7704  .8904 

1.4428 

40 

30 

0.1309 

.1305   .1157 

.9914  .9963 

.1317   .1194 

7.5958  .8806 

1.4399 

30 

40 

a  1338 

.1334   .1252 

.991  1  .9961 

.1346   .1291 

7.4287  .8709 

1.4370 

20 

50 

0.1367 

.1363   .1345 

.9907  .9959 

.1376   .1385 

7.2687  .8615 

1.4341 

10 

8°  00' 

0.1396 

.1392  9.1436 

.9903  9.9958 

.1405  9.1478 

7.1154  0.8522 

1.4312 

82°  00' 

10 

0.1425 

.1421   .1525 

.9899  .9956 

.1435   .1569 

6.9682  .8431 

1.4283 

50 

20 

0.1454 

.1449   .1612 

.9894  .9954 

.1465   .1658 

6.8269  .8342 

1.4254 

40 

30 

0.1484 

.1478   .1697 

.9890  .9952 

.1495   .1745 

6.6912  .8255 

1.4224 

30 

40 

0.1513 

.1507   .1781 

.9886  .9950 

.1524   .1831 

6.5606  .8169 

1.4195 

20 

50 

0.1542 

.1536   .1863 

.9881  .9948 

.1554   .1915 

6.4348  .8085 

1.4166 

10 

9°  (XX 

0.1571 

.1564  9.1943 

.9877  9.9946 

.1584  9.1997 

6.3138  0.8003 

1.4137 

81°  00' 

Nat.  Log.* 

Nat.  Log.* 

Nat.  Log.* 

Nat.   Log. 

Cosines 

Sines 

Cotangents 

Tangents 

Ra- 
dians 

De- 
grees 

MATHEMATICAL  TABLES 


53 


TRIGONOMETRIC  FUNCTIONS     (continued) 

Annex  -10 in  columns  marked*.     (For  O.°l  intervals,  see  pp.  46-51) 


De- 
grees 

Ra- 
dians 

Sines 

Cosines 

Tangents 

Cotangent 

Nat.    Log.* 

Nat.   Log.* 

Nat.   Log.* 

Nat.    Log. 

9°  00' 

0.1571 

.1564    9.1943 

.9877    9.9946 

.1584    9.1997 

6.3138  0.8003 

1.4137 

81"  (XX 

kio 

0.1600 

.1593      .2022 

.9872      .9944 

.1614      .2078 

6.1970    .7922 

1.4108 

50 

20 

0.1629 

.1622      .2100 

.9868      .9942 

.1644      .2158 

6.0844    .7842 

1.4079 

40 

30 

0.1658 

.1650      .2176 

.9863      .9940 

.1673      .2236 

5.9758    .7764 

1.4050 

30 

40 

0.1687 

.1679      .2251 

.9858      .9938 

.1703      .2313 

5.8708    .7687 

1.4021 

20 

50 

0.1716 

.1708      .2324 

.9853      .9936 

.1733      2389 

5.7694    .7611 

1.3992 

10 

10°  00' 

0.1745 

.1736    9.2397 

.9848    9.9934 

.1763    9.2463 

5.6713  0.7537 

1.3963 

80°  (XX 

10 

0.1774 

.1765      .2468 

.9843      .9931 

.1793      2536 

5.5764    .7464 

1.3934 

50 

20 

0.1804 

.1794      .2538 

.9838      .9929 

.1823      2609 

5.4845    .7391 

1.3904 

40 

30 

0.1833 

.1822      .2606 

.9833      .9927 

.1853      2680 

5.3955    .7320 

1.3875 

30 

40 

0.1862 

.1851       .2674 

.9827      .9924 

.1883      2750 

5.3093   .7250 

1.3846 

20 

50 

0.1891 

.1880      .2740 

.9822      .9922 

.1914      2819 

5.2257    .7181 

1.3817 

10 

11°  00' 

0.1920 

.1908    9.2806 

.9816    9.9919 

.1944    92887 

5.1446  0.7113 

1.3788 

79°0(X 

10 

0.1949 

.1937      .2870 

.9811      .9917 

.1974      2953 

5.0658    .7047 

1.3759 

50 

20 

0.1978 

.1965      .2934 

.9805      .9914 

.2004      .3020 

4.9894    .6980 

1.3730 

40 

30 

0.2007 

.1994      .2997 

.9799      .9912 

2035      .3085 

4.9152    .6915 

1.3701 

30 

40 

0.2036 

.2022      .3058 

.9793      .9909 

2065      .3149 

4.8430    .6851 

1.3672 

20 

50 

0.2065 

•2051      .3119 

.9787      .9907 

2095      .3212 

4.7729    .6788 

1.3643 

10 

12°  00' 

0.2094 

.2079    9.3179 

.9781     9.9904 

2126    9.3275 

4.7046  0.6725 

1.3614 

78°  00' 

10 

0.2123 

.2108      .3238 

.9775      .9901 

2156      3336 

4.6382    .6664 

1.3584 

50 

20 

0.2153 

.2136      .3296 

.9769      .9899 

2186      .3397 

4.5736    .6603 

1.3555 

40 

30 

0.2182 

.2164      .3353 

.9763      .9896 

2217      .3458 

4.5107    .6542 

1.3526 

30 

40 

0.2211 

.2193      .3410 

.9757      .9893 

2247      .3517 

4.4494    .6483 

1.3497 

20 

50 

0.2240 

.2221      .3466 

.9750      .9890 

2278      .3576 

4.3897    .6424 

1.3468 

10 

13°  00' 

0.2269 

.2250    9.3521 

.9744    9.9887 

2309    9.3634 

4.3315  0.6366 

1.3439 

77°  (XX 

10 

0.2298 

.2278      .3575 

.9737      .9884 

2339      .3691 

4.2747    .6309 

1.3410 

50 

20 

0.2327 

.2306      .3629 

.9730      .9881 

2370      .3748 

4.2193    .6252 

1.3381 

40 

30 

0.2356 

.2334      .3682 

.9724      .9878 

2401      .3804 

4.1653    .6196 

1.3352 

30 

40 

0.2385 

.2363      .3734 

.9717      .9875 

2432      .3859 

4.1126    .6141 

1.3323 

20 

50 

0.2414 

.2391      .3786 

.9710      .9872 

2462      .3914 

4.0611     .6086 

1.3294 

10 

14°  00' 

0.2443 

.2419    9.3837 

.9703    9.9869 

2493    9.3968 

4.0108  0.6032 

1.3265 

76°  (XX 

10 

0.2473 

.2447      .3887 

.9696      .9866 

2524      .4021 

3.9617    .5979 

1.3235 

50 

20 

0.2502 

.2476      .3937 

.9689      .9863 

2555      .4074 

3.9136    .5926 

1.3206 

40 

30 

0.2531 

2504      .3986 

.9681      .9859 

2586      .4127 

3.8667    .5873 

1.3177 

30 

40 

0.2560 

2532      .4035 

.9674      .9856 

2617      .4178 

3.8208    .5822 

1.3148 

20 

50 

0.2589 

2560      .4083 

.9667      .9853 

2648      .4230 

3.7760    .5770 

13119 

10 

15°  00' 

0.2618 

.2588    9.4130 

.9659    9.9849 

2679    9.4281 

3.7321  0.5719 

1.3090 

75°  00' 

10 

0.2647 

2616      .4177 

.9652      .9846 

2711      .4331 

3.6891     .5669 

1.3061 

50 

20 

0.2676 

2644      .4223 

.9644      .9843 

.2742      .4381 

3.6470    .5619 

1.3032 

40 

30 

0.2705 

2672      .4269 

.9636    -.9839 

.2773      .4430 

3.6059    .5570 

1.3003 

30 

40 

0.2734 

2700      .4314 

.9628      .9836 

2805      .4479 

3.5656    .5521 

1.2974 

20 

50 

0.2763 

2728      .4359 

.9621      .9832 

2836      -4527 

3.5261     .5473 

U945 

10 

16°  00' 

0.2793 

2756    9.4403 

.9613    9.9828 

.2867    9.4575 

3.4874  0.5425 

12915 

74°  (XX 

10 

0.2822 

2784      .4447 

.9605      .9825 

2899      .4622 

3.4495    .5378 

12886 

50 

20 

0.2851 

2812      .4491 

^9596      .9821 

2931      .4669 

3.4124    .5331 

1.2857 

40 

30 

0.2880 

2840      .4533 

.9588      .9817 

2962      .4716 

3.3759    .5284 

1.2828 

30 

40 

0.2909 

2868      .4576 

.9580      .9814 

2994      .4762 

3.3402    .5238 

12799 

20 

50 

0.2938 

2896      .4618 

.9572      .9810 

.3026      .4808 

3.3052    .5192 

12770 

10 

17°  (XX 

0.2967 

2924    9.4659 

.9563    9.9806 

.3057    9.4853 

3.2709  0.5147 

1.2741 

3°  (XX 

10 

0.2996 

2952      .4700 

.9555      .9802 

3089      .4898 

3.2371     .5102 

12712 

50 

20 

0.3025 

2979      .4741 

.9546      .9798 

3121       .4943 

3.2041     .5057 

1.2683 

40 

30 

0.3054 

3007      .4781 

9537      .9794 

3153      .4987 

3.1716    .5013 

1.2654 

30 

40 

0.3083 

3035      .4821 

9528      .9790 

3185      .5031 

3.1397    .4969 

1.2625 

20 

50 

0.3113 

3062      .4861 

9520      .9786 

3217      .5075 

3.1084    .4925 

12595 

10 

18°  (XX 

0.3142 

3090    9.4900 

9511    9.9782 

3249    9.5118 

3.0777  0.4882 

12566 

2°  (XX 

Nat.    Log.* 

Nat.    Log.* 

Nat.   Log.* 

Nat.    Log. 

Cosines 

Sines 

Cotangents 

Tangents 

Ra- 
dians 

De- 
grees 

54 


MATHEMATICAL   TABLES 


TRIGONOMETRIC  FUNCTIONS 

Annex— 10 in  columns  marked*. 


(continued) 

(For  O.°l  intervals,  Bee  pp.  46-51) 


De- 
grees 

Ra- 
dians 

Sines 

Cosines 

Tangents 

Cotangents 

Nat.  Log.  * 

Nat.  Log.* 

Nat.  Log.* 

Nat.  Log. 

18°  00' 

0.3142 

.3090  9.4900 

.9511  9.9782 

.3249  9.5118 

3.0777  0.4882 

1.2566 

72°  00* 

10 

0.3171 

.3118   .4939 

.9502   .9778 

.3281   .5161 

3.0475  .4839 

1.2537 

50 

20 

0.3200 

.3145   .4977 

.9492   .9774 

.3314   .5203 

3.0178  .4797 

1.^508 

40 

30 

0.3229 

.3173   .5015 

.9483   .9770 

,3346   .5245 

2.9887  .4755 

1.2479 

30 

40 

0.3258 

.3201   .5052 

.9474   .9765 

.3378   .5287 

2.9600  .4713 

1  .2450 

20 

50 

0.3287 

.3228   .5090 

.9465   .9761 

.3411   .5329 

2.9319  .4671 

1.2421 

10 

19°  00' 

0.3316 

.3256  9.5126 

.9455  9.9757 

.3443  9.5370 

2.9042  0.4630 

1.2392 

71°  00' 

10 

0.3345 

.3283   .5163 

.9446   .9752 

.3476   3411 

2.8770  .4589 

1.2363 

50 

20 

0.3374 

.3311   .5199 

.9436   .9748 

.3508   .5451 

2.8502  .4549 

1  .2334 

40 

30 

0.3403 

.3338   .5235 

.9426   .9743 

.3541   .5491 

2.8239  .4509 

1.2305 

30 

40 

0.3432 

.3365   .5270 

.9417   .9739 

.3574   .5531 

2.7980  .4469 

.2275 

20 

50 

0.3462 

.3393   .5306 

.9407   .9734 

.3607   .5571 

2.7725  .4429 

12246 

10 

20°  00' 

0.3491 

.3420  9.5341 

.9397  9.9730 

.3640  9.5611 

2.7475  0.4389 

1.2217 

70°  00' 

10 

0.3520 

.3448   .5375 

.9387   .9725 

.3673   .5650 

2.7228  .4350 

1.2188 

50 

20 

0.3549 

3475   .5409 

.9377   .9721 

.3706   .5689 

2.6985  .4311 

1.2159 

40 

30 

0.3578 

.3502   .5443 

.9367   .9716 

.3739   .5727 

2.6746  .4273 

1.2130 

30 

40 

0.3607 

.3529   .5477 

.9356   .971  1 

.3772   .5766 

2.6511  .4234 

1.2101 

20 

50 

0.3636 

.3557   .5510 

.9346   .9706 

.3805   .5804 

2.6279  .4196 

1.2072 

10 

21°  00' 

0.3665 

.3584  9.5543 

.9336  9.9702 

.3839  9.5842 

2.6051  0.4158 

1.2043 

69°  0(X 

10 

0.3694 

.361  1   .5576 

.9325   .9697 

.3872   .5879 

2.5826  .4121 

1.2014 

50 

20 

0.3723 

.3638   .5609 

.9315   .9692 

.3906   .5917 

2.5605  .4083 

.1985 

40 

30 

0.3752 

.3665   .5641 

.9304   .9687 

.3939   .5954 

2.5386  .4046 

.1956 

30 

40 

0.3782 

.3692   .5673 

.9293   .9682 

.3973   .5991 

2.5172  .4009 

.1926 

20 

50 

0.381  1 

.3719   .5704 

.9283   .9677 

.4006   .6028 

2.4960  .3972 

.1897 

10 

22°  00' 

0.3840 

.3746  9.5736 

.9272  9.9672 

.4040  9.6064 

2.4751  0.3936 

.1868 

68°  00' 

10 

0.3869 

.3773   .5767 

.9261   .9667 

.4074   .6100 

2.4545  .3900 

.1839 

50 

20 

0.3898 

.3800   .5798 

.9250   .9661 

.4108   .6136 

2.4342  .3864 

.1810 

40 

30 

0.3927 

.3827   .5828 

.9239   .9656 

.4142   .6172 

2.4142  .3828 

.1781 

30 

40 

0.3956 

.3854   .5859 

.9228   .9651 

.4176   .6208 

2.3945  .3792 

.1752 

20 

50 

0.3985 

.3881   .5889 

.9216   .9646 

.4210   .6243 

2.3750  .3757 

.1723 

10 

23°  00' 

0.4014 

.3907  9.5919 

.9205  9.9640 

.4245  9.6279 

2.3559  0.3721 

.1694 

67°  00' 

10 

0.4043 

.3934   .5948 

.9194   .9635 

.4279   .6314 

2.3369  .3686 

.1665 

50 

20 

0.4072 

.3961   .5978 

.9182   .9629 

.4314   .6348 

2.3183  .3652 

.1636 

40 

30 

0.4102 

.3987   .6007 

.9171   .9624 

.4348   .6383 

2.2998  .3617 

.1606 

30 

30 

0.4131 

.4014   .6036 

.9159   .9618 

.4383   .6417 

2.2817  .3583 

.1577 

20 

50 

0.4160 

.4041   .6065 

.9147   .9613 

.4417   .6452 

2.2637  .3548 

.1548 

10 

24°  00' 

0.4189 

.4067  9.6093 

.9135  9.9607 

.4452  9.6486 

2.2460  0.3514 

.1519 

66°  00' 

10 

0.4218 

.4094   .6121 

.9124   .9602 

.4487   .6520 

2.2286  .3480 

.1490 

50 

20 

0.4247 

.4120   .6149 

.9112   .9596 

.4522   .6553 

2.2113  .3447 

.1461 

40 

30 

0.4276 

.4147   .6177 

.9100   .9590 

.4557   .6587 

2.1943  .3413 

.1432 

30 

40 

0.4305 

.4173   .6205 

.9088   .9584 

.4592   .6620 

2.1775  .3380 

.1403 

20 

50 

0.4334 

.4100   .6232 

.9075   .9579 

.4628   .6654 

2.1609  .3346 

.1374 

10 

25°  0(K 

0.4363 

.4226  9.6259 

.9063  9.9573 

.4663  9.6687 

2.1445  0.3313 

.1345 

65°  00' 

10 

0.4392 

.4253   .6286 

.9051   .9567 

.4699   .6720 

2.1283  .3280 

.1316 

50 

20 

0.4422 

.4279   .6313 

.9038   .9561 

.4734   .6752 

2.1123  .3248 

.1286 

40 

30 

0.4451 

.4305   .6340 

.9026   .9555 

.4770   .6785 

2.0965  .3215 

.1257 

30 

40 

0.4480 

.4331   .6366 

.9013   .9549 

.4806   .6817 

2.0809  .3183 

.1228 

20 

50 

0.4509 

.4358   .6392 

.9001   .9543 

.4841   .6850 

2.0655  .3150 

.1199 

10 

26°  00' 

0.4538 

.4384  9.6418 

.8988  9.9537 

.4877  9.6882 

2.0503  0-3118 

.1170 

64°  00' 

10 

0.4567 

.4410   .6444 

.8975   .9530 

.4913   .6914 

2.0353  .3086 

.1141 

50 

20 

0.4596 

.4436   .6470 

.8962   .9524 

.4950   .6946 

2.0204  .3054 

.1112 

40 

30 

0.4625 

.4462   .6495 

.8949   .9518 

.4986   .6977 

2.0057  .3023 

.1083 

30 

40 

0.4654 

.4488   .6521 

.8936   .9512 

.5022   .7009 

1.9912  .2991 

.1054 

20 

50 

0.4683 

.4514  .6546 

.8923  .9505 

.5059  .7040 

1.9768  .2960 

.1025 

10 

27°  (XX 

0.4712 

.4540  9.6570 

.8910  9.9499 

.5095  9.7072 

1.9626  0.2928 

1.0996 

63°  00' 

Nat.  Log.* 

Nat.  Log.* 

Nat.  Log.* 

Nat.  Log. 

Cosines 

Sines 

Cotangents 

Tangents 

Ra- 
dians 

De- 
grees 

MATHEMATICAL   TABLES 


55 


TRIGONOMETRIC  FUNCTIONS     (continued) 

Annex  -10  in  columns  marked*.     (For  0°.l  intervals,  see  pp.  46-51) 


De- 
grees 

Ra- 
dians 

Sines 

Cosines 

Tangents 

Cotangents 

Nat.  Log.* 

Nat.  Log.* 

Nat.  Log.* 

Nat.  Log. 

27°  00' 

0.4712 

.4540  9.6570 

.8910  9.9499 

.5095  9.7072 

1.9626  0.2928 

1.0996 

63°  00' 

10 

0.4741 

.4566   .6595 

.8897   .9492 

.5132   .7103 

1.9486  2897 

1.0966 

50 

20 

0.4771 

.4592   .6620 

.8884   .9486 

.5169   .7134 

1.9347  .2866 

1.0937 

40 

30 

0.4800 

.4617   .6644 

.8870   .9479 

.5206   .7165 

1.9210  .2835 

1.0908 

30 

40 

0.4829 

.4643   .6668 

.8857   .9473 

.5243   .7196 

1.9074  .2804 

1.0879 

20 

50 

0.4858 

.4669   .6692 

.8843   .9466 

.5280   .7226 

1.8940  .2774 

1.0850 

10 

28°  00' 

0.4887 

.4695  9.6716 

.8829  9.9459 

.5317  9.7257 

1.8807  0.2743 

1.0821 

62°  00' 

10 

0.4916 

.4720   .6740 

.8816   .9453 

.5354   .7287 

1.8676  .2713 

1.0792 

50 

20 

0.4945 

.4746   .6763 

.8802   .9446 

.5392   .7317 

1.8546  .2683 

1.0763 

40 

30 

0.4974 

.4772   .6787 

.8788   .9439 

.5430   .7348 

1.8418  .2652 

1.0734 

30 

40 

0.5003 

.4797   .6810 

.8774   .9432 

.5467   .7378 

1.8291  .2622 

1.0705 

20 

50 

0.5032 

.4823   .6833 

.8760   .9425 

.5505   .7408 

1.8165  .2592 

1.0676 

10 

29°  00' 

0.5061 

.4848  9.6856 

.8746  9.9418 

.5543  9.7438 

1.8040  0.2562 

1.0647 

61°  00' 

10 

0.5091 

.4874   .6878 

.8732   .941  1 

.5581   .7467 

1.7917  .2533 

1.0617 

50 

20 

0.5120 

.4899   .690! 

.8718   .9404 

.5619   .7497 

1.7796  .2503 

1.0588 

40 

30 

0.5149 

.4924   .6923 

.8704   .9397 

.5658   .7526 

1.7675  .2474 

1.0559 

30 

40 

0.5178 

.4950   .6946 

.8689   .9390 

.5696   .7556 

1.7556  .2444 

1.0530 

20 

50 

0.5207 

.4975   .6968 

.8675   .9383 

.5735   .7585 

1.7437  .2415 

1.0501 

10 

30°  00' 

0.5236 

.5000  9.6990 

.8660  9.9375 

.5774  9.7614 

1.7321  0.2386 

1.0472 

60°  00' 

10 

0.5265 

.5025   .7012 

.8646   .9368 

.5812   .7644 

1.7205  .2356 

1.0443 

50 

20 

0.5294 

.5050   .7033 

.8631   .9361 

.5851   .7673 

1.7090  .2327 

1.0414 

40 

30 

0.5323 

.5075   .7055 

.8616   .9353 

.5890   .7701 

1.6977  .2299 

1.0385 

30 

40 

0.5352 

.5100   .7076 

.8601   .9346 

.5930   .7730 

1.6864  .2270 

1.0356 

20 

50 

0.5381 

.5125   .7097 

.8587   .9338 

.5969   .7759 

1.6753  .2241 

1.0327 

10 

31°  00' 

0.5411 

.5150  9.7118 

.8572  9.9331 

.6009  9.7788 

1.6643  0.2212 

1.0297 

59°  00' 

10 

0.5440 

.5175   .7139 

.8557   .9323 

.6048   .7816 

1.6534  .2184 

1.0268 

50 

20 

0.5469 

.5200   .7160 

.8542   .9315 

.6088   .7845 

1.6426  .2155 

1.0239 

40 

30 

0.5498 

.5225   .7181 

.8526   .9308 

.6128   .7873 

1.6319  .2127 

1.0210 

30 

40 

0.5527 

.5250   .7201 

.8511   .9300 

.6168   .7902 

1.6212  .2098 

1.0181 

20 

50 

0.5556 

.5275   .7222 

.8496   .9292 

.6208   .7930 

1.6107  .2070 

1.0152 

10 

32°  00' 

0.5585 

.5299  9.7242 

.8480  9.9284 

.6249  9.7958 

.6003  0.2042 

1.0123 

58°  00' 

10 

0.5614 

.5324   .7262 

.8465   .9276 

.6289   .7986 

.5900  .2014 

1.0094 

50 

20 

0.5643 

.5348   .7282 

.8450   .9268 

.6330   .8014 

.5798  .1986 

1  .0065 

40 

30 

0.5672 

.5373   .7302 

.8434   .9260 

.6371   .8042 

.5697  .1958 

1.0036 

30 

40 

0.5701 

.5398   .7322 

.8418   .9252 

.6412   .8070 

.5597  .1930 

1.0007 

20 

50 

0.5730 

5422   .7342 

.8403   .9244 

.6453   .8097 

.5497  .1903 

0.9977 

10 

33°  (XX 

0.5760 

5446  9.7361 

.8387  9.9236 

.6494  9.8125 

.5399  0.1875 

0.9948 

57°  00' 

10 

0.5789 

5471   .7380 

.8371   .9228 

.6536   .8153 

.5301  .1847 

0.9919 

50 

20 

0.5818 

5495   .7400 

.8355   .9219 

.6577   .8180 

.5204  .1820 

0.9890 

40 

30 

0.5847 

5519   .7419 

.8339   .9211 

.6619   .8208 

.5108  .1792 

0.9861 

30 

40 

0.5876 

5544  .7438 

.8323   .9203 

.6661   .8235 

.5013  .1765 

0.9832 

20 

50 

0.5905 

5568  .7457 

.8307   .9194 

.6703   .8263 

.4919  .1737 

0.9803 

10 

34°  00' 

0.5934 

5592  9.7476 

.8290  9.9186 

.6745  9.8290 

.4826  0.1710 

09774 

56°  00' 

10 

0.5963 

5616   .7494 

.8274   .9177 

.6787   .8317 

.4733  .1683 

0.9745 

50 

20 

0.5992 

5640   .7513 

.8258   .9169 

.6830   .8344 

.4641  .1656 

0.9716 

40 

30 

0.6021 

5664   .7531 

.8241   .9160 

.6873   .8371 

.4550  .1629 

0.9687 

30 

40 

0.6050 

5688   .7550 

.8225   .9151 

.6916   .8398 

.4460  .1602 

0.9657 

20 

50 

0.6080 

5712   .7568 

.8208   .9142 

.6959   .8425 

.4370  .1575 

0.9628 

10 

35°  00' 

0.6109 

5736  9.7586 

.8192  9.9134 

.7002  9.8452 

.4281  0.1548 

0.9599 

55°  00' 

10 

0.6138 

5760   .7604 

.8175   .9125 

.7046   .8479 

.4193  .1521 

0.9570 

50 

20 

0.6167 

5783   .7622 

.8158   .9116 

.7089   .8506 

.4106  .1494 

0.9541 

40 

30 

0.6196 

5807   .7640 

.8141   .9107 

.7133   .8533 

.4019  .1467 

0.9512 

30 

40 

0.6225 

5831   .7657 

.8124   .9098 

.7177   .8559 

.3934  .1441 

0.9483 

20 

50 

0.6254 

5854   .7675 

.8107   .9089 

.7221   .8586 

.3848  .1414 

0.9454 

10 

36°  00' 

0.6283 

5878  9.7692 

.8090  9.9080 

.7265  9.8613 

1.3764  0.1387 

0.9425 

54°  00' 

Nat.  Log.* 

Nat.  Log.* 

Nat.  Log.* 

Nat.  Log. 

Cosines 

Sines 

Cotangents 

Tangents 

Ra- 
dians 

De- 
grees 

56 


MATHEMATICAL  TABLES 


TRIGONOMETRIC  FUNCTIONS      (continued) 

Annex  -10  in  columns  marked*.     (For  0°.l  intervals,  see  pp.  4.6-51) 


De- 
grees 

Ra- 
dians 

Sines 

Cosines 

Tangents 

Cotangents 

Nat.  Log.* 

Nat.  Log.* 

Nat.  Log.* 

Nat.  Log. 

36°  (XX 

0.6283 

3878  9.7692 

.8090  9.9080 

.7265  9.8613 

1.3764  0.1387 

0.9425 

54°  00' 

10 

0.6312 

.5901   .7710 

.8073   .9070 

.7310   .8639 

1.3680  .1361 

0.9396 

50 

20 

0.6341 

.5925   .7727 

.8056   .9061 

.7355   .8666 

1.3597  .1334 

0.9367 

40  ' 

30 

0.6370 

.5948   .7744 

.8039   .9052 

.7400   .8692 

1.3514  .1308 

0.9338 

30 

40 

0.6400 

.5972   .7761 

.8021   .9042 

.7445   .8718 

1.3432  .1282 

0.9308 

20 

50 

0.6429 

.5995   .7778 

.8004   .9033 

.7490   .8745 

1.3351  .1255 

0.9279 

10 

37°  00' 

0.6458 

.6018  9.7795 

.7986  9.9023 

.7536  9.8771 

.3270  0.1229 

0.9250 

53°  00' 

10 

0.6487 

.6041   .781  1 

.7969   .9014 

.7581   .8797 

.3190  .1203 

0.9221 

50 

20 

0.6516 

.6065   .7828 

.7951   .9004 

.7627   .8824 

.3111  .1176 

0.9192 

40 

30 

0.6545 

.6088   .7844 

.7934   .8995 

.7673   .8850 

.3032  .1150 

0.9163 

30 

40 

0.6574 

.61  1  1   .7861 

.7916   .8985 

.7720   .8876 

.2954  .1124 

0.9134 

20 

50 

0.6603 

.6134   .7877 

.7898   .8975 

.7766   .8902 

.2876  .1098 

0.9105 

•  10 

38°  00' 

0.6632 

.6157  9.7893 

.7880  9.8965 

.7813  9.8928 

.2799  0.1072 

0.9076 

52°  00' 

10 

0.6661 

.6180   .7910 

.7862   .8955 

.7860   .8954 

.2723  .1046 

0.9047 

50 

20 

0.6690 

.6202   .7926 

.7844   .8945 

.7907   .8980 

.2647  .1020 

0.9018 

40 

30 

0.6720 

.6225   .7941 

.7826   .8935 

.7954   .9006 

.2572  .0994 

0.8988 

30 

40 

0.6749 

.6248   .7957 

.7808   .8925 

.8002   .9032 

.2497  .0968 

0.8959 

20 

50 

0.6778 

.6271   .7973 

.7790   .8915 

.8050   .9058 

.2423  .0942 

0.8930 

10 

39°  00' 

0.6807 

.6293  9.7989 

.7771  9.8905 

.8098  9.9084 

.2349  0.0916 

0.8901 

51°  00' 

10 

0.6836 

.6316   .8004 

.7753   .8895 

.8146   .9110 

.2276  .0890 

0.8872 

50 

20 

0.6865 

.6338   .8020 

.7735   .8884 

.8195   .9135 

.2203  .0865 

0.8843 

40 

30 

0.6894 

.6361   .8035 

.7716   .8874 

.8243   .9161 

.2131  .0839 

0.8814 

30 

40 

0.6923 

.6383   .8050 

.7698   .8864 

.8292   .9187 

.2059  .0813 

0.8785 

20 

50 

0.6952 

.6406   .8066 

.7679   .8853 

.8342   .9212 

.1988  .0788 

0.8756 

10 

40°  00' 

0.6981 

.6428  9.8081 

.7660  9.8843 

.8391  9.9238 

.1918  0.0762 

0.8727 

50°  00' 

10 

0.7010 

.6450   .8096 

.7642   .8832 

.8441   .9264 

.1847  .0736 

0.8698 

50 

20 

0.7039 

.6472   .81  1  1 

.7623   .8821 

.8491   .9289 

.1778  .0711 

0.8668 

40 

30 

0.7069 

.6494   .8125 

.7604   .8810 

.8541   .9315 

.1708  .0685 

0.8639 

30 

40 

0.7098 

.6517   .8140 

.7585   .8800 

.8591   .9341 

.1640  .0659 

0.8610 

20 

50 

0.7127 

.6539   .8155 

.7566   .8789 

.8642   .9366 

.1571  .0634 

0.8581 

10 

41°  00' 

0.7156 

.6561  9.8169 

.7547  9.8778 

.8693  9.9392 

.1504  0.0608 

0.8552 

49°  00' 

10 

0.7185 

.6583   .8184 

.7528   .8767 

.8744   .9417 

J436  .0583 

0.8523 

50 

20 

0.7214 

.6604   .8198 

.7509   .8756 

.8796   .9443 

.1369  .0557 

0.8494 

40 

30 

0.7243 

.6626   .8213 

.7490   .8745 

.8847   .9468 

.1303  .0532 

0.8465 

30 

40 

0.7272 

.6648   .8227 

.7470   .8733 

.8899   .9494 

.1237  .0506 

0.8436 

20 

50 

0.7301 

.6670   .8241 

.7451   .8722 

.8952   .9519 

.1171  .0481 

0.8407 

10 

42°  00' 

0.7330 

.6691  9.8255 

.7431  9.8711 

.9004  9.9544 

.1106  00456 

0.8378 

48°  00' 

10 

0.7359 

.6713   .8269 

.7412   .8699 

.9057   .9570 

.1041  .0430 

0.8348 

50 

20 

0.7389 

.6734   .8283 

.7392   .8688 

.9110   .9595 

1  .0977  .0405 

0.8319 

40 

30 

0.7418 

.6756   .8297 

.7373   .8676 

.9163   .9621 

1.0913  .0379 

0.8290 

30 

40 

0.7447 

.6777   .8311 

.7353   .8665 

.9217   .9646 

1.0850  .0354 

0.8261 

20 

50 

0.7476 

.6799   .8324 

.7333   .8653 

.9271   .9671 

1.0786  .0329 

0.8232 

10 

43°  00' 

0.7505 

.6820  9.8338 

.7314  9.8641 

.9325  9.9697 

1.0724  0.0303 

0.8203 

47°  00' 

10 

0.7534 

.6841   .8351 

.7294   .8629 

.9380   .9722 

1.0661  .0278 

0.8174 

50 

20 

0.7563 

.6862   .8365 

.7274   .8618 

.9435   .9747 

1.0599  .0253 

0.8145 

40 

30 

0.7592 

.6884   .8378 

.7254   .8606 

.9490   .9772 

1.0538  .0228 

0.8116 

30 

40 

0.7621 

.6905   .8391 

.7234   .8594 

.9545   .9798 

1.0477  .0202 

0.8087 

20 

50 

0.7650 

.6926   .8405 

.7214   .8582 

.9601   .9823 

1.0416  .0177 

0.8058 

10 

44°  00' 

0.7679 

.6947  9.8418 

.7193  9.8569 

.9657  9.9848 

1.0355  0.0152 

0.8029 

46°  00' 

10 

0.7709 

.6967   .8431 

.7173   .8557 

.9713   .9874 

1.0295  .0126 

0.7999 

50 

20 

0.7738 

.6988   .8444 

.7153   .8545 

.9770   .9899 

1.0235  .0101 

0.7970 

40 

30 

0.7767 

.7009   .8457 

.7133   .8532 

.9827   .9924 

1.0176  .0076 

0.7941 

30 

40 

07796 

.7030   .8469 

.7112   .8520 

.9884   .9949 

1.0117  .0051 

0.7912 

20 

50 

0.7825 

.7050   .8482 

.7092   .8507 

.9942   .9975 

1.0058  .0025 

0.7883 

10 

45°  00' 

0.7854 

.7071  9.8495 

.7071  9.8495 

1.0000  0.0000 

1.0000  0.0000 

0.7854 

45°  00' 

Nat.  Log.* 

Nat.  Log.* 

Nat.  Log.* 

Nat.  Log. 

Cosines 

Sines 

Cotangents 

Tangents 

Ra- 
dians 

De- 
grees 

MATHEMATICAL   TABLES 
EXPONENTIALS     [e»  and  <T"] 


57 


n 

n           *H 

c    s 

n 

,  g 

n 

en 

n 

<-   % 

n 

c~" 

n 

e~* 

0.00 

1.000    in 

0.50 

1.649    ,, 

1.0 

2.718* 

0.00 

1.000      ]0 

0.50 

.607 

1.0 

.368* 

.01 

i.oto    5 

.51 

1.665     S 

.1 

3.004 

.01 

0.990-  J 

31 

.600 

.1 

.333 

.02 

J.020      2 

.52 

1.682      i 

.2 

3.320 

.02 

.980  ~  2 

.52 

.595 

2 

.301 

.03 

1.030      9 

.53 

1.699    \l 

3 

3.669 

.03 

.970  -'g 

.53 

.589 

3 

.273 

.04 

1.041     JJ 

.54 

1.716    \77 

.4 

4.055 

.04 

•961  I,J 

.54 

.583 

.4 

.247 

0.05 

1.051     .. 

0.55 

1.733    1R 

1.5 

4.482 

0.05 

.951        „ 

0.55 

.577 

13 

.223 

.06 

.062 

.56 

1.751      2 

.6 

4.953 

.06 

.942  ~,J 

.56 

.571 

.    .6 

202 

.07 

1.073      A 

.57 

1.768    \i 

.7 

5.474 

.07 

.932  ~'§ 

.57 

.566 

.7 

.183 

.08 

1.083      ? 

.58 

1.786      2 

.8 

6.050 

.08 

.923  ~  I 

.58 

.560 

.8 

.165 

.09 

1.094    jj 

.59 

1.804    !« 

.9 

6.686 

.09 

.914  I  99 

39 

.554 

.9 

.150 

0.10 

.105    ,. 

0.60 

1.822    ,. 

2.0 

7.389 

0.10 

.905       o 

0.60 

.549 

2.0 

.135 

.11 

.116 

.61 

1.840 

.1 

8.166 

.11 

.896  ~  A 

.61 

.543 

.1 

.122 

.12 

.127      I 

.62 

1.859      A 

2 

9.025 

.12 

.887  ~  I 

.62 

.538 

2 

.111 

.13 

.139      2 

.63 

1.878    \l 

3 

9.974 

.13 

.878  ~  J 

.63 

.533 

3 

.100 

.14 

.150    jj 

.64 

1.896    JJ 

.4 

11.02 

.14 

.869  ~  I 

.64 

.527 

.4 

.0907 

0.15 

.162    ,, 

0.65 

1.916    ]0 

2.5 

12.18 

0.15 

.861        Q 

0.65 

.522 

23 

.0821 

.16 

.174      2, 

.66 

1.935    \l 

.6 

13.46 

.16 

.852  ~  I 

.66 

.517 

.6 

.0743 

.17 

.185      ' 

.67 

1.954    £ 

.7 

14.88 

.17 

.844  ~  I 

.67 

.512 

.7 

.0672 

.18 

.197      2 

.68 

1.974    % 

.8 

16.44 

.18 

.835  ~  I 

.68 

.507 

.8 

.0608 

.19 

1.209    j| 

.69 

1.994    20 

.9 

18.17 

.19 

.827  I  I 

.69 

.502 

.9 

.0550 

0.20 

1.221     „ 

0.70 

2.014    7n 

3.0 

20.09 

0.20 

.819       » 

0.70 

.497 

3.0 

.0498 

.21 

1.234    \l 

.71 

2.034    ?,S 

.1 

22.20 

.21 

.811  ~ 

.71 

.492 

.1 

.0450 

.22 

1.246      2 

.72 

2.054    20 

.2 

24.53 

.22 

.803  - 

.72 

.487 

.2 

.0408 

.23 

1.259      2 

.73 

2.075    2, 

.3 

27.11 

.23 

.795  ~  5 

.73 

.482 

3 

.0369 

24 

1.271     !| 

.74 

2.096    21 

.4 

29.% 

.24 

-787  I  I 

.74 

.477 

.4 

.0334 

0.25 

1.284    n 

0.75 

2.117    „ 

3.5 

33.12 

0.25 

.779       o 

0.75 

.472 

33 

.0302 

.26 

707      13 

.^y/    |  o 

.76 

2.138    ?! 

.6 

36.60 

.26 

.771  ~  I 

.76 

.468 

.6 

.0273 

.27 

1.310      ^ 

.77 

2.160    ?f 

.7 

40.45 

.27 

.763  ~  5 

.77 

.463 

.7 

.0247 

.28 

1.323      | 

.78 

2.181     tl 

.8 

44.70 

.28 

.756  ~  I 

.78 

.458 

.8 

.0224 

.29 

1.336    JJ 

.79 

2.203    g 

.9 

49.40 

.29 

.748  ~  J 

.79 

.454 

.9 

.0202 

O.SO 

1.350    n 

0.80 

2.226    22 

4.0 

54.60 

0.30 

.741        „ 

0.80 

.449 

4.0 

.0183 

.31 

1.363    \\ 

.81 

2.248    £ 

.1 

60.34 

.31 

.733  ~  2 

.81 

.445 

.1 

.0166 

.32 

1.377 

.82 

2.270    S2, 

.2 

66.69 

.32 

.726  ~  I 

.82 

.440 

.2 

.0150 

.33 

1.391      J 

.83 

2.293    g 

3 

73.70 

.33 

.719  ~  I 

.83 

.436 

.3 

.0136 

.34 

1.405    ]< 

.84 

2.3.6    g 

.4 

81.45 

.34 

.7.2  ~  ] 

.84 

.432 

.4 

.0123 

0.35 

1.419    14 

0.85 

2.340    21 

4.5 

90.02 

0.35 

.705       7 

0.85 

.427 

4.5 

.0111 

.36 

1.433      i 

.86 

2.363    5? 

.36 

.698  ~  i 

.86 

.423 

.37 

1.448 

.87 

2.387    Si 

5.0 

148.4 

.37 

.691  ~  I 

.87 

.419 

5.0 

.00674 

.38 

1.462      J 

.88 

2.411    2J 

6.0 

403.4 

.38 

.684  ~  i 

.88 

.415 

6.0 

.00248 

.39 
0.40 

1.477    jf 

1.492      ,e 

.89 
0.90 

2.435    24 
2.460    94 

7.0 
8.0 

1097. 
2981. 

.39 
0.40 

.677  I  ij 
.670 

.89 
0.90 

.411 
.407 

8.0 

i.  000912 
.000335 

.41 

1.507      1 

.91 

2.484    24 

9.0 

8103. 

.41 

.664"  S 

.91 

.403 

9.0 

.000123 

.42 

1.522    \\ 

.92 

2.509    25 

10.0 

22026. 

.42 

.657  "  I 

.92 

.399 

10.0 

.000045 

.43 
.44 

1.537      5 
1.553    Jf 

.93 
.94 

2.535    fS 

•y  <Mn    ** 

2.560    26 

«/2 
fcr/2 

4.810 
23.14 

.43 
.44 

.651  "  S 
•644  I  ?6 

.93 
.94 

.395 
.391 

IT/2 
27T/2 

.208 
.0432 

0.45 

1.568    ,, 

0.95 

2.586    7, 

37T/2 

111.3 

0.45 

.638       7 

0.95 

.387 

3r/2 

.00898 

.46 

1.584     9 

.96 

2.612    g 

47T/2 

535.5 

.46 

.631  "  ? 

.96 

.383 

fcr/2 

.00187 

.47 

1.600 

.97 

2.638    ?9 

5T/2 

2576. 

.47 

.625  " 

.97 

.379 

5r/2 

.000388 

.48 

1.616    JJ 

.98 

2.664    $ 

for/2 

12392. 

.48 

.619  ~  9 

.98 

.375 

6r/2 

.000081 

.49 

1.632      2 

.99 

2.691    g 

lir/2 

59610. 

.49 

.613  ~  I 

.99 

.372 

7^/2 

.000017 

0.50 

I/ 

1.649 

1.00 

// 
2.718 

8^/2 

286751. 

0.50 

0.607 

1.00 

368 

W2 

.000003 

*NOTE:  Do  not  interpolate  in  this  column. 

e  =  2.71828         1/e  =  0.367879       logioe  =  0.4343        1/(0.4343)  -  2.3026 

logio(0.4343)  =  1.6378        logio(e»)  =  n(0.4343) 

For  table  of  multiples  of  0.4343,  see  p.  62.    Graphs,  p.  174. 


58 


MATHEMATICAL  TABLES 


HYPERBOLIC  LOGARITHMS 


n 

n  (2.3026) 

n  (0.6974-3) 

These  two  pages  give  the  natural  (hyper- 
bolic,   or    Napierian)    logarithms    (log«)   of 
numbers  between  1  and  10,  correct  to  four 

1 
2 
3 
4 

2.3026 
4.6052 
6.9078 
9.2103 

0.6974-3 
0.3948-5 
0.0922-7 
0.7897-10 

places.     Moving  the  decimal  point  n  places 

5 

11.5129 

0.4871-12 

to  the  right  [or  left]  in  the  number  is  equiva- 

6 

13.8155 

0.1845-14 

lent  to  adding  n  times  2.3026  [or  n  times 
3.6974]  to  the  logarithm.  Base  e  =  2.71828  + 

; 

8 
9 

16.1181 
18.4207 
20.7233 

08819-17 
0.5793-19 
0.2767-21 

EU- 

& 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

£ 

1.0 

0.0000 

0100 

0198 

0296 

0392 

0488 

0583 

0677 

0770 

0862 

95 

1.1 

0953 

1044 

1133 

1222 

1310 

1398 

1484 

1570 

1655 

1740 

87 

1.2 

1823 

1906 

1989 

2070 

2151 

2231 

2311 

2390 

2469 

2546 

80 

1.3 

2624 

2700 

2776 

2852 

2927 

3001 

3075 

3148 

3221 

3293 

74 

1.4 

3365 

3436 

3507 

3577 

3646 

3716 

3784 

3853 

3920 

3988 

69 

1.5 

0.4055 

4121 

4187 

4253 

4318 

4383 

4447 

4511 

4574 

4637 

65 

1.6 

4700 

4762 

4824 

4886 

4947 

5008 

5068 

5128 

5188 

5247 

61 

1.7 

5306 

5365 

5423 

5481 

5539 

5596 

5653 

5710 

5766 

5822 

57 

1.8 

5878 

5933 

5988 

6043 

6098 

6152 

6206 

6259 

6313 

6366 

54 

1.9 

6419 

6471 

6523 

6575 

6627 

6678 

6729 

6780 

6831 

6881 

51 

2.0 

0.6931 

6981 

7031 

7080 

7129 

7178 

7227 

7275 

7324 

7372 

49 

2.1 

7419 

7467 

7514 

7561 

7608 

7655 

7701 

7747 

7793 

7839 

47 

2.2 

7885 

7930 

7975 

8020 

8065 

8109 

8154 

8198 

8242 

8286 

44 

2.3 

8329 

8372 

8416 

8459 

8502 

8544 

8587 

8629 

8671 

8713 

43 

2.4 

8755 

8796 

8838 

8879 

8920 

8961 

9002 

9042 

9083 

9123 

41 

2.5 

0.9163 

9203 

9243 

9282 

9322 

9361 

9400 

9439 

9478 

9517 

39 

2.6 

9555 

9594 

9632 

9670 

9708 

9746 

9783 

9821 

9858 

9895 

38 

2.7 

0.9933 

9969 

*0006 

*0043 

*0080 

*0116 

*0152 

*OI88 

*0225 

*0260 

36 

2.8 

1.0296 

0332 

0367 

0403 

0438 

0473 

0508 

0543 

0578 

0613 

35 

2.9 

0647 

0682 

0716 

0750 

0784 

0818 

0852 

0886 

0919 

0953 

34 

3.0 

1.0986 

1019 

1053 

1086 

1119 

1151 

1184 

1217 

1249 

1282 

33 

3.1 

1314 

1346 

1378 

1410 

1442 

1474 

1506 

1537 

1569 

1600 

32 

3.2 

1632 

1663 

1694 

1725 

1756 

1787 

1817 

1848 

1878 

1909 

31 

3.3 

1939 

1969 

2000 

2030 

2060 

2090 

2119 

2149 

2179 

2208 

30 

3.4 

2238 

2267 

2296 

2326 

2355 

2384 

2413 

2442 

2470 

2499 

29 

3.5 

1.2528 

2556 

2585 

2613 

2641 

2669 

2698 

2726 

2754 

2782 

28 

3.6 

2809 

2837 

2865 

2892 

2920 

2947 

2975 

3002 

3029 

3056 

27 

3.7 

3083 

3110 

3137 

3164 

3191 

3218 

3244 

3271 

3297 

3324 

27 

3.8 

3350 

3376 

3403 

3429 

3455 

3481 

3507 

3533 

3558 

3584 

26 

3.9 

3610 

3635 

3661 

3686 

3712 

3737 

3762 

3788 

3813 

3838 

25 

4.0 

1.3863 

3888 

3913 

3938 

3962 

3987 

4012 

4036 

4061 

4085 

25 

4.1 

4110 

4134 

4159 

4183 

4207 

4231 

4255 

4279 

4303 

4327 

24 

4.2 

4351 

4375 

4398 

4422 

4446 

4469 

4493 

4516 

4540 

4563 

23 

4.3 

4586 

4609 

4633 

4656 

4679 

4702 

4725 

4748 

4770 

4793 

23 

4.4 

4816 

4839 

4861 

4884 

4907 

4929 

4951 

4974 

4996 

5019 

22 

4.5 

1.5041 

5063 

5085 

5107 

5129 

5151 

5173 

5195 

5217 

5239 

22 

4.6 

5261 

5282 

5304 

5326 

5347 

5369 

5390 

5412 

5433 

5454 

21 

4.7 

5476 

5497 

5518 

5539 

5560 

5581 

5602 

5623 

5644 

5665 

21 

4.8 

5686 

5707 

5728 

5748 

5769 

5790 

5810 

5831 

5851 

5872 

20 

4.9 

5892 

5913 

5933 

5953 

5974  v 

5994 

6014 

6034 

6054 

6074 

20 

logtX  =  (2.3026)  logio  x  logioa:  =  (0.4343)  Iog8  x 

where  2.3026  =•  log«w  and  0.4343  =>  logioe  (see  p.  62).     For  graph*,  see  p.  174. 


MATHEMATICAL  TABLES 


59 


HYPERBOLIC  LOGARITHMS   (continued} 


e^ 

3  <o 
53-° 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

!i 

5.0 

1.6094 

6114 

6134 

6154 

6174 

6194 

6214 

6233 

6253 

6273 

20 

5.1 

6292 

6312 

6332 

6351 

6371 

6390 

6409 

6429 

6448 

6467 

19 

5.2 

6487 

6506 

6525 

6544 

6563 

6582 

6601 

6620 

6639 

6658 

19 

5.3 

6677 

6696 

6715 

6734 

6752 

6771 

6790 

6808 

6827 

6845 

18 

5.4 

6864 

.6882 

6901 

6919 

6938 

6956 

6974 

6993 

7011 

7029 

18 

5.5 

1.7047 

7066 

7084 

7102 

7120 

7138 

7156 

7174 

7192 

7210 

18 

5.6 

7228 

7246 

7263 

7281 

7299 

7317 

7334 

7352 

7370 

7387 

18 

5.7 

7405 

7422 

7440 

7457 

7475 

7492 

7509 

7527 

7544 

7561 

17 

5.8 

7579 

7596 

7613 

7630 

7647 

7664 

7681 

7699 

7716 

7733 

17 

5.9 

7750 

7766 

7783 

7800 

7817 

7834 

7851 

7867 

7884 

7901 

17 

6.0 

1.7918 

7934 

7951 

7967 

7984 

8001 

8017 

8034 

8050 

8066 

16 

6.1 

8083 

8099 

8116 

8132 

8148 

8165 

8181 

8197 

8213 

8229 

16 

6.2 

8245 

8262 

8278 

8294 

8310 

8326 

8342 

8358 

8374 

8390 

16 

6.3 

8405 

8421 

8437 

8453 

8469 

8485 

8500 

8516 

8532 

8547 

16 

6.4 

8563 

8579 

8594 

8610 

8625 

8641 

8656 

8672 

8687 

8703 

15 

6.5 

1.8718 

8733 

8749 

8764 

8779 

8795 

8810 

8825 

8840 

8856 

15 

6.6 

8871 

8886 

8901 

8916 

8931 

8946 

8961 

8976 

8991 

9006 

15 

6.7 

9021 

9036 

9051 

9066 

9081 

9095 

9110 

9125 

9140 

9155 

15 

6.8 

9169 

9184 

9199 

9213 

9228 

9242 

9257 

9272 

9286 

9301 

15 

6.9 

9315 

9330 

9344 

9359 

9373 

9387 

9402 

9416 

9430 

9445 

14 

7.0 

1.9459 

9473 

9488 

9502 

9516 

9530 

9544 

9559 

9573 

9587 

14 

7.1 

9601 

9615 

9629 

9643 

9657 

9671 

9685 

9699 

9713 

9727 

14 

7.2 

9741 

9755 

9769 

9782 

9796 

9810 

9824 

9838 

9851 

9865 

14 

7.3 

1  .9879 

9892 

9906 

9920 

9933 

9947 

9961 

9974 

9988 

*OOOI 

13 

7.4 

2.0015 

0028 

0042 

0055 

0069 

0082 

0096 

0109 

0122 

0136 

13 

7.5 

2.0149 

0162 

0176 

0189 

0202 

0215 

0229 

0242 

0255 

0268 

13 

7.6 

0281 

0295 

0308 

0321 

0334 

0347 

0360 

0373 

0386 

0399 

13 

7.7 

0412 

0425 

0438 

0451 

0464 

0477 

0490 

0503 

0516 

0528 

13 

7.8 

0541 

0554 

0567 

0580 

0592 

0605 

0618 

0631 

0643 

0656 

13 

7.9 

0669 

0681 

0694 

0707 

0719 

0732 

0744 

0757 

0769 

0782 

12 

8.0 

2.0794 

.  0807 

0819 

0832 

0844 

0857 

0869 

0882 

0894 

0906 

12 

8.1 

0919 

0931 

0943 

0956 

0968 

0980 

0992 

1005 

1017 

1029 

12 

8.2 

1041 

1054 

1066 

1078 

1090 

1102 

1114 

1126 

1138 

1150 

12 

8.3 

1163 

1175 

1187 

1199 

1211 

1223 

1235 

1247 

1258 

1270 

12 

8.4 

1282 

1294 

1306 

1318 

1330 

1342 

1353 

1365 

1377 

1389 

12 

8.5 

2.1401 

1412 

1424 

1436 

1448 

1459 

1471 

1483 

1494 

1506 

12 

8.6 

1518 

1529 

1541 

1552 

1564 

1576 

1587 

1599 

1610 

1622 

12 

8.7 

1633 

1645 

1656 

1668 

1679 

1691 

1702 

1713 

1725 

1736 

11 

8.8 

1748 

1759 

1770 

1782 

1793 

1804 

1815 

1827 

1838 

1849 

It 

8.9 

1861 

1872 

1883 

1894 

1905 

1917 

1928 

1939 

1950 

1961 

11 

9.0 

2.1972 

1983 

1994 

2006 

2017 

2028 

2039 

2050 

2061 

2072 

11 

9.1 

2083 

2094 

2105 

2116 

2127 

2138 

2148 

2159 

2170 

2181 

II 

9.2 

2192 

2203 

2214 

2225 

2235 

2246 

2257 

2268 

2279 

2289 

11 

9.3 

2300 

2311 

2322 

2332 

2343 

2354 

2364 

2375 

2386 

2396 

11 

9.4 

2407 

2418 

2428 

2439 

2450 

2460 

2471 

2481 

2492 

2502 

11 

9.5 

2.2513 

2523 

2534 

2544 

2555 

2565 

2576 

2586 

2597 

2607 

10 

9.6 

2618 

2628 

2638 

2649 

2659 

2670 

2680 

2690 

2701 

2711 

10 

9.7 

2721 

2732 

2742 

2752 

2762 

2773 

2783 

2793 

2803 

2814 

10 

9.8 

2824 

2834 

2844 

2854 

2865 

2875 

2885 

2895 

2905 

2915 

10 

9.9 

2925 

2935 

2946 

2956 

2966 

2976 

2986 

29% 

3006 

3016 

10 

10.0 

2.3026 

Moving  the  decimal  point  n  places  to  the  right  [or  left]  in  the  number  requires  adding 
n  times  2.3026  for  n  times  (0.6974-3)]  in  the  body  of  the  table.  See  auxiliary  table  of 
multiples  on  top  of  the  preceding  page. 


60  MATHEMATICAL  TABLES 

HYPERBOLIC  SINES  [sinh  x  =  ^(e*  -  e~*)] 


X 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

£ 

0.0 

.0000 

.0100 

.0200 

.0300 

.0400 

.0500 

.0600 

.0701 

.0801 

.0901 

00 

1 

.1002 

.1102 

.1203 

.1304 

.1405 

.1506 

.1607 

.1708 

.1810 

.1911 

01 

2 

.2013 

.2115 

.2218 

.2320 

.2423 

.2526 

.2629 

.2733 

.2837 

.2941 

03 

3 

.3045 

.3150 

.3255 

.3360 

.3466 

.3572 

.3678 

.3785 

.3892 

.4000 

106 

4 

.4108 

.4216 

.4325 

.4434 

.4543 

.4653 

.4764 

.4875 

.4986 

.5098 

no 

0.5 

.5211 

.5324 

.5438 

.5552 

.5666 

.5782 

.5897 

.6014 

.6131 

.6248 

116 

6 

.6367 

.6485 

.6605 

.6725 

.6846 

.6967 

.7090 

.7213 

.7336 

.7461 

122 

7 

.7586 

.7712 

.7838 

.7966 

.8094 

.8223 

.8353 

.8484 

.8615 

.8748 

130 

8 

.8881 

.9015 

.9150 

.9286 

.9423 

.9561 

.9700 

.9840 

.9981 

1.012 

138 

9 

1.027 

1.041 

.055 

1.070 

1.085 

1.099 

1.114 

1.129 

1.145 

1.160 

15 

1.0 

1.175 

1.191 

.206 

1.222 

1.238 

1.254 

1.270 

1.286 

1.303 

1.319 

16 

1 

1.336 

1.352 

.369 

1.386 

1.403 

1.421 

1.438 

1.456 

1.474 

1.491 

17 

2 

1.509 

1.528 

.546 

1.564 

1.583 

1.602 

1.621 

1.640 

1.659 

1.679 

19 

3 

1.698 

1.718 

.738 

1.758 

1.779 

•    1.799 

1.820 

1.841 

1.862 

1.883 

21 

4 

1.904 

1.926 

.948 

1.970 

1.992 

ZOI4 

2.037 

2.060 

2.083 

2.106 

22 

1.5 

2.129 

2.153 

2.177 

2.201 

2.225 

2.250 

2.274 

2.299 

2.324 

2.350 

25 

6 

2.376 

2.401 

2.428 

2.454 

2.481 

2.507 

2.535 

2.562 

2.590 

2.617 

27 

7 

2.646 

2.674 

2.703 

2.732 

2.761 

2.790 

2.820 

2.850 

2.881 

2.911 

30 

8 

2.942 

2.973 

3.005 

3.037 

3.069 

3.101 

3.134 

3.167 

3.200 

3.234 

33 

9 

3.268 

3.303 

3.337 

3.372 

3.408 

3.443 

3.479 

3.516 

3.552 

3.589 

36 

2.0 

3.627 

3.665 

3.703 

3.741 

3.780 

3.820 

3.859 

3.899 

3.940 

3.981 

39 

1 

4.022 

4.064 

4.106 

4.148 

4.191 

4.234 

4.278 

4.322 

4.367 

4.412 

44 

2 

4.457 

4.503 

4.549 

4.596 

4.643 

4.691 

4.739 

4.788 

4.837 

4.887 

48 

3 

4.937 

4.988 

5.039 

5.090 

5.142 

5.195 

5.248 

5.302 

5.356 

5.411 

53 

4 

5.466 

5.522 

5.578 

5.635 

5.693 

5.751 

5.810 

5.869 

5.929 

5.989 

58 

2.5 

6.050 

6.112 

6.174 

6.237 

6.300 

6.365 

6.429 

6.495 

6.561 

6.627 

64 

6 

6.695 

6.763 

6.831 

6.901 

6.971 

7.042 

7.113 

7.185 

7.258 

7.332 

71 

7 

7.406 

7.481 

7.557 

7.634 

7.711 

7.789 

7.868 

7.948 

8.028 

8.110 

79 

8 

8.192 

8.275 

8.359 

8.443 

8.529 

8.615 

8.702 

8.790 

8.879 

8.969 

87 

9 

9.060 

9.151 

9.244 

9.337 

9.431 

9.527 

9.623 

9.720 

9.819 

9.918 

96 

3.0 

10.02 

10.12 

10.22 

10.32 

10.43 

10.53 

10.64 

10.75 

10.86 

10.97 

11 

1 

11.08 

11.19 

11.30 

11.42 

11.53 

11.65 

11.76 

11.88 

12.00 

12.12 

12 

2 

12.25 

12.37 

12.49 

12.62 

12.75 

12.88 

13.01 

13.14 

13.27 

13.40 

13 

3 

13.54 

13.67 

13.81 

13.95 

14.09 

14.23 

14.38 

14.52 

14.67 

14.82 

14 

4 

14.97 

15.12 

15.27 

15.42 

15.58 

15.73 

15.89 

16.05 

16.21 

16.38 

16 

3.5 

16.54 

16.71 

16.88 

17.05 

17.22 

17.39 

17.57 

17.74 

17.92 

18.10 

17 

6 

18.29 

18.47 

18.66 

18.84 

19.03 

19.22 

19.42 

19.61 

19.81 

20.01 

19 

7 

20.21 

20.41 

20.62 

20.83 

21.04 

21.25 

21.46 

21.68 

21.90 

22.12 

21 

8 

22.34 

22.56 

22.79 

23.02 

23.25 

23.49 

23.72 

23.96 

24.20 

24.45 

24 

9 

24.69 

24.94 

25.19 

25.44 

25.70 

25.96 

26.22 

26.48 

26.75 

27.02 

26 

4.0 

27.29 

27.56 

27.84 

28.12 

28.40 

28.69 

28.98 

29.27 

29.56 

29.86 

29 

1 

30.16 

30.47 

30.77 

31.08 

31.39 

31.71 

32.03 

32.35 

32.68 

33.00 

32 

2 

33.34 

33.67 

34.01 

34.35 

34.70 

35.05 

35.40 

35.75 

36.11 

36.48 

35 

3 

36.84 

37.21 

37.59 

37.97 

38.35 

38.73 

39.12 

39.52 

39.91 

40.31 

39 

4 

40.72 

41.13 

41.54 

41.96 

42.38 

42.81 

43.24 

43.67 

44.11 

44.56 

43 

4.5 

45.00 

45.46 

45.91 

46.37 

46.84 

47.31 

47.79 

48.27 

48.75 

49.24 

47 

6 

49.74 

5024 

50.74 

51.25 

51.77 

52.29 

52.81 

53.34 

53.88 

54.42 

52 

7 

54.97 

55.52 

56.08 

56.64 

57.21 

57.79 

58.37 

58.96 

59.55 

60.15 

58 

8 

60.75 

61.36 

61.98 

62.60 

63.23 

63.87 

64.51 

65.16 

65.81 

66.47 

64 

9 

67.14 

67.82 

68.50 

69.19 

69.88 

70.58 

71.29 

72.01 

72.73 

73.46 

71 

5.0 

74.20 

If  x  >  5,  sinh  x  =  W(e*)  and  logio  sinh  x  =  (0.4343)z  +  0.6990  —  1,  correct  to  four 
significant  figures.     For  table  of  multiples  of  0.4343,  see  p.  62.     Graphs,  p.  174. 


MATHEMATICAL  TABLES  61 

HYPERBOLIC  COSINES  [cosh  x  =  K(e*  +e~*)] 


V 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

£ 

0.0 

1.000 

1.000 

1.000 

1.000 

1.001 

1.001 

1.002 

1.002 

1.003 

1.004 

1 

1 

.005 

1.006 

1.007 

1.008 

1.010 

1.011 

1.013 

1.014 

1.016 

1.018 

2 

2 

.020 

1.022 

1.024 

1.027 

1.029 

1.031 

1.034 

1.037 

1.039 

1.042 

3 

3 

.045 

1.048 

.052 

1.055 

1.058 

1.062 

1.066 

1.069 

1.073 

1.077 

4 

4 

.081 

1.085 

.090 

1.094 

1.098 

1.103 

1.108 

1.112 

1.117 

1.122 

5 

0.5 

.128 

1.133 

.138 

1.144 

1.149 

1.155 

1.161 

1.167 

1.173 

1.179 

6 

6 

.185 

1.192 

.198 

1.205 

1.212 

1.219 

1.226 

1.233 

1.240 

1.248 

7 

7 

.255 

1.263 

.271 

1.278 

1.287 

1.295 

1.303 

1.311 

1.320 

1.329 

8 

8 

.337 

1.346 

.355 

1.365 

1.374 

1.384 

1.393 

1.403 

1.413 

1.423 

10 

9 

1.433 

1.443 

.454 

1.465 

1.475 

1.486 

1.497 

1.509 

1.520 

1.531 

11 

1.0 

1.543 

1.555 

1.567 

1.579 

1.591 

1.604 

1.616 

1.629 

1.642 

1.655 

13 

1 

1.669 

1.682 

1.696 

1.709 

1.723 

1.737 

1.752 

1.766 

1.781 

1.796 

14 

2 

1.811 

1.826 

1.841 

1.857 

1.872 

1.888 

1.905 

1.921 

1.937 

1.954 

16 

3 

1.971 

1.988 

2.005 

2.023 

2.040 

2.058 

2.076 

2.095 

2.113 

2.132 

18 

4 

2.151 

2.170 

2.189 

2.209 

2.229 

2.249 

2.269 

2.290 

2.310 

2.331 

20 

1.5 

2.352 

2.374 

2.395 

2.417 

2.439 

2.462 

2.484 

2.507 

2.530 

2.554 

23 

6 

2.577 

2.601 

2.625 

2.650 

2.675 

2.700 

2.725 

2.750 

2.776 

2.802 

25 

7 

2.828 

2.855 

2.882 

2.909 

2.936 

2.964 

2.992 

3.021 

3.049 

3.078 

28 

8 

3.107 

3.137 

3167 

3.197 

3.228 

3.259 

3.290 

3.321 

3.353 

3.385 

31 

9 

3.418 

3.451 

3.484 

3.517 

3.551 

3.585 

3.620 

3.655 

3.690 

3.726 

34 

2.0 

3.762 

3.799 

3.835 

3.873 

3.910 

3.948 

3.987 

4.026 

4.065 

4.104 

38 

1 

4.144 

4.185 

4.226 

4.267 

4.309 

4.351 

4.393 

4.436 

4.480 

4.524 

42 

2 

4.568 

4.613 

4.658 

4.704 

4.750 

4.797 

4.844 

4.891 

4.939 

4.988 

47 

3 

5.037 

5.087 

5.137 

5.188 

5.239 

5.290 

5.343 

5.395 

5.449 

5.503 

52 

4 

5.557 

5.612 

5.667 

5.723 

5.780 

5.837 

5.895 

5.954 

6.013 

6.072 

58 

2.5 

6.132 

6.193 

6.255 

6.317 

6.379 

6.443 

6.507 

6.571 

6.636 

6.702 

64 

6 

6.769 

6.836 

6.904 

6.973 

7.042 

7.112 

7.183 

7.255 

7.327 

7.400 

70 

7 

7.473 

7.548 

7.623 

7.699 

7.776 

7.853 

7.932 

8.011 

8.091 

8.171 

78 

8 

8.253 

8.335 

8.418 

8.502 

8.587 

8.673 

8.759 

8.847 

8.935 

9.024 

86 

9 

9.115 

9.206 

9.298 

9.391 

9.484 

9.579 

9.675 

9.772 

9.869 

9.968 

95 

3.0 

10.07 

10.17 

10.27 

10.37 

10.48 

10.58 

10.69 

10.79 

10.90 

11.01 

11 

1 

11.12 

11.23 

11.35 

11.46 

11.57 

11.69 

11.81 

11.92 

12.04 

12.16 

12 

2 

12.29 

12.41 

12.53 

12.66 

12.79 

12.91 

13.04 

13.17 

13.31 

13.44 

13 

3 

13.57 

13.71 

13.85 

13.99 

14.13 

14.27 

14.41 

14.56 

14.70 

14.85 

14 

4 

15.00 

15.15 

15.30 

15.45 

15.ftl 

15.77 

15.92 

16.08 

16.25 

16.41 

16 

3.5 

16.57 

16.74 

16.91 

17.08 

17.25 

17.42 

17.60 

17.77 

17.95 

18.13 

17 

6 

18.31 

18.50 

18.68 

18.87 

19.06 

19.25 

19.44 

19.64 

19.84 

20.03 

19 

7 
8 

20.24 
22.36 

20.44 
22.59 

20.64 
22.81 

20.85 
23.04 

21.06 
23.27 

21.27 
23.51 

21.49 
23.74 

21.70 
23.98 

21.92 
24.22 

22.14 
24.47 

21 
23 

9 

24.71 

24.96 

25.21 

25.46 

25.72 

25.98 

26.24 

26.50 

26.77 

27.04 

26 

4.0 

27.31 

27.58 

27.86 

28.14 

28.42 

28.71 

29.00 

29.29 

29.58 

29.88 

29 

1 

30.18 

30.48 

30.79 

31.10 

31.41 

31.72 

32.04 

32.37 

32.69 

33.02 

32 

2 

33.35 

33.69 

34.02 

34.37 

34.71 

35.06 

35.41 

35.77 

36.13 

36.49 

35 

3 

36.86 

37.23 

37.60 

37.98 

38.36 

38.75 

39.13 

39.53 

39.93 

40.33 

39 

4 

40.73 

41.14 

41.55 

41.97 

42.39 

42.82 

43.25 

43.68 

44.12 

44.57 

43 

4.E 

45.01 

45.47 

45.92 

46.38 

46.85 

47.32 

47.80 

48.28 

48.76 

49.25 

47 

6 

49.75 

50.25 

50.75 

51.26 

51.78 

52.30 

52.82 

53.35 

53.89 

54.43 

52 

7 

54.98 

55.53 

56.09 

56.65 

57.22 

57.80 

58.38 

58.96 

59.56 

60.15 

58 

8 

60.76 

61.37 

61.99 

62.61 

63.24 

63.87 

64.52 

65.16 

65.82 

66.48 

64 

9 

67.15 

67.82 

68.50 

69.19 

69.89 

70.59 

71.30 

72.02 

72.74 

73.47 

71 

5.0 

74.21 

If  x  >  5,  cosh  x  =  ^i(e*)  and  logio  cosh  x  =  (0.4343)*  +  0.6990  —  1,  correct  to  four  signifi- 
cant figures.    For  table  of  multiples  of  0.4343,  see  p.  62.     Graphs,  p.  174. 


62  MATHEMATICAL  TABLES 

HYPEEBOLIC  TANGENTS  [tanh  x  =  (e*-e~*) /(e*  +<T*)  =  sinh  z/cosh 


X 

0 

1 

2 

3 

4 

5 

6 

(1 

8 

9 

|l 

0.0 

.0000 

.0100 

.0200 

.0300 

.0400 

.0500 

.0599 

.0699 

.0798 

.0898 

100 

.0997 

.1096 

.1194 

.1293 

.1391 

.1489 

.1587 

.1684 

.1781 

.1878 

98 

2 

.1974 

.2070 

.2165 

.2260 

.2355 

.2449 

.2543 

.2636 

.2729 

.2821 

94 

3 

.2913 

.3004 

.3095 

.3185 

.3275 

.3364 

.3452 

.3540 

.3627 

.3714 

89 

4 

.3800 

.3885 

.3969 

.4053 

.4136 

.4219 

.4301 

.4382 

.4462 

.4542 

82 

0.5 

.4621 

.4700 

.4777 

.4854 

.4930 

.5005 

.5080 

.5154 

.5227 

.5299 

75 

6 

.5370 

.5441 

.5511 

.5581 

.5649 

.5717 

.5784 

.5850 

.5915 

.5980 

67 

7 

.6044 

.6107 

.6169 

.6231 

.6291 

.6352 

.6411 

.6469 

.6527 

.6584 

60 

8 

.6640 

.6696 

.6751 

.6805 

.6858 

.6911 

.6963 

.7014 

.7064 

.7114 

52 

9 

.7163 

.7211 

.7259 

.7306 

.7352 

.7398 

.7443 

.7487 

.7531 

.7574 

45 

1.0 

.7616 

.7658 

.7699 

.7739 

.7779 

.7818 

.7857 

.7895 

.7932 

.7969 

39 

1 

.8005 

.8041 

.8076 

.8110 

.8144 

.8178 

.8210 

.8243 

.8275 

.8306 

33 

2 

.8337 

.8367 

.8397 

.8426 

.8455 

.8483 

.8511 

.8538 

.8565 

.8591 

28 

3 

.8617 

.8643 

.8668 

.8693 

.8717 

.8741 

.8764 

.8787 

.8810 

.8832 

24 

4. 

.8854 

.8875 

.8896 

.8917 

.8937 

.8957 

.8977 

.8996 

.9015 

.9033 

20 

1.5 

.9052 

.9069 

.9087 

.9104 

.9121 

.9138 

.9154 

.9170 

.9186 

.9202 

17 

6 

.9217 

.9232 

.9246 

.9261 

.9275 

.9289 

.9302 

.9316 

.9329 

.9342 

14 

7 

.9354 

.9367 

.9379 

.9391 

.9402 

.9414 

.9425 

.9436 

.9447 

.9458 

11 

8 

.9468 

.9478 

.9488 

.9498 

.9508 

.9518 

.9527 

.9536 

.9545 

.9554 

9 

9 

.9562 

.9571 

.9579 

.9587 

.9595 

.9603 

.9611 

.9619 

.9626 

.9633 

8 

2.0 

.9640 

.9647 

.9654 

.9661 

.9668 

.9674 

.9680 

.9687 

.9693 

.9699 

6 

1 

.9705 

.9710 

.9716 

.9722 

.9727 

.9732 

.9738 

.9743 

.9748 

.9753 

5 

2 

.9757 

.9762 

.9767 

.9771 

.9776 

.9780 

.9785 

.9789 

.9793 

.9797 

4 

3 

.9801 

.9805 

.9809 

.9812 

.9816 

.9820 

.9823 

.9827 

.9830 

.9834 

4 

4 

.9837 

.9840 

.9843 

.9846 

.9849 

.9852 

.9855 

.9858 

.9861 

.9863 

3 

2.5 

.9866 

.9869 

.9871 

.9874 

.9876 

.9879 

.9881 

.9884 

.9886 

.9888 

2 

6 

.9890 

.9892 

.9895 

.9897 

.9899 

.9901 

.9903 

.9905 

.9906 

.9908 

2 

7 

.9910 

.9912 

.9914 

.9915 

.9917 

.9919 

.9920 

.9922 

.9923 

.9925 

2 

8 

.9926 

.9928 

.9929 

.9931 

.9932 

.9933 

.9935 

.9936 

.9937 

.9938 

2.9 

.9940 

.9941 

.9942 

.9943 

.9944 

.9945 

.9946 

.9947 

.9949 

.9950 

1 

3. 

.9951 

.9959 

.9967 

.9973 

.9978 

.9982 

.9985 

.9988 

.9990 

.9992 

4 

4. 

.9993 

.9995 

.9996 

.9996 

.9997 

.9998 

.9998 

.9998 

.9999 

.9999 

I 

5. 

.9999 

If  x  >  5, 

tanh  a; 

=  1.0000  to  four  decimal  places.  Graphs,  p 

.  174. 

MULTIPLES  OF  0.4343     (0.43429448  =  logw  e) 


X 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

0. 

0.0000 

0.0434 

0.0869 

0.1303 

0.1737 

0.2171 

0.2606 

0.3040 

0.3474 

0.390< 

1. 

0.4343 

0.4777 

0.5212 

0.5646 

0.6080 

0.6514 

0.6949 

0.7383 

0.7817 

0.825: 

2. 

0.8686 

0.9120 

0.9554 

0.9989 

1.0423 

1.0857 

1.1292 

1.1726 

1.2160 

1  .259! 

3. 

1  .3029 

1.3463 

1.3897 

1.4332 

1.4766 

1.5200 

1.5635 

1.6069 

1.6503 

1  .6933 

4. 

1.7372 

1.7806 

1.8240 

1.8675 

1.9109 

1.9543 

1.9978 

2.0412 

2.0846 

2.128( 

5. 

2.1715 

2.2149 

2.2583 

2.3018 

2.3452 

2.3886 

2.4320 

2.4755 

2.5189 

2.5622 

6. 

2.6058 

2.6492 

2.6926 

2.7361 

2.7795 

2.8229 

2.8663 

2.9098 

2.9532 

2.996* 

7. 

3.0401 

3.0835 

3.1269 

3.1703 

3.2138 

3.2572 

3.3006 

3.3441 

3.3875 

3.43TC 

8. 

3.4744 

3.5178 

3.5612 

3.6046 

3.6481 

3.6915 

3.7349 

3.7784 

3.8218 

3.8652 

9. 

3.9087 

3.9521 

3.9955 

4.0389 

4.0824 

4.1258 

4.1692 

4.2127 

4.2561 

4.299* 

MULTIPLES  OP 

2.3026 

(2.3025851  = 

1/0.4343) 

x 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

0. 

0.0000 

0.2303 

0.4605 

0.6908 

0.9210 

1.1513 

1.3816 

1.6118 

1.8421 

2.0723 

1. 

2.3026 

2.5328 

2.7631 

2.9934 

3.2236 

3.4539 

3.6841 

3.9144 

4.1447 

4.3749 

2. 

4.6052 

4.8354 

5.0657 

5.2959 

5.5262 

5.7565 

5.9867 

6.2170 

6.4472 

6.6775 

3. 

6.9078 

7.1380 

7.3683 

7.5985 

7.8288 

8.0590 

8.2893 

8.5196 

8.7498 

8.9801 

4. 

9.2103 

9.4406 

9.6709 

9.901  1 

10.131 

10.362 

10.592 

10.822 

11.052 

11.283 

5. 

11.513 

11.743 

11.973 

12.204 

12.434 

12.664 

12.894 

13.125 

13.355 

13.585 

6. 

13.816 

14.046 

14.276 

14.506 

14.737 

14.967 

15.197 

15.427 

15.658 

15.888 

7. 

16.118 

'     16.348 

16.579 

16.809 

17.039 

17.269 

17.500 

17.730 

17.960 

18.190 

8. 

18.421 

18.651 

18.881 

19.111 

19.342 

19.572 

19.802 

20.032 

20.263 

20.493 

9. 

20.723 

20.954 

21.184 

21.414 

21.644 

21.875 

22.105 

22.335 

22.565 

22.796 

MATHEMATICAL  TABLES 


63 


STANDARD 
DISTRIBUTION  OF 
RESIDUALS  (p.  121) 

a  =  any   positive   quantity; 
y  =  the     number     of    residuals 
which  are  numerically  <  a; 
r  =  the  probable  error  of  a  single 
observation; 

n  «=  number  of  observations. 

a 

y 

Diff. 

r 

n 

0.0 

.000 

1 
2 

.054 
.107 

54 
53 

CO 

3 
4 

.160 
.213 

JJ 

53 
51 

0.5 
6 
7 

.264 
.314 
.363 

50 
49 

8 
9 

.411 
.456 

45 
44 

1.0 

.500 

1 

.542 

^/ 

2 
3 

.582 
.619 

40 
37 

4 

.655 

33 

1.5 
6 
7 
8 
9 

.688 
.719 
.748 
.775 
.  .800 

31 
29 
27 
25 
23 

2.0 

1 

.823 

.843 

20 

i  (i 

2 

.862 

1  7 

3 

4 

.879 
.895 

17 
16 

13 

2.5 

.908 

6 

.921 

7 
8 

.931 
.941 

10 
10 

9 

.950 

9 
7 

3.0 

.957 

1 

.963 

6 

2 

.969 

6 

3 

.974 

5 

4 

.978 

4 

4 

3.5 

.982 

6 

.985 

3 

7 

.987 

8 

.990 

3 

9 

.991 

2 

4.0 

.993 

6 

5.0 

.999 

FACTORS  FOR  COMPUTING 
PROBABLE  ERROR  (p.  121) 


n 

Bessel 

Peters 

0.6745 

0.6745 

0.8453 

0.8453 

V(n  -  1) 

Vn(n-l) 

Vn(/i-l) 

n\/n—  1 

2 

.6745 

.4769 

.5978 

.4227 

3 

.4769 

.2754 

.3451 

.1993 

4 

.3894 

.1947 

.2440 

.1220 

5 

.3372 

.1508 

.1890 

.0845 

6 

.3016 

.1231 

.1543 

.0630 

7 

.2754 

.1041 

.1304 

.0493 

8 

.2549 

.0901 

.1130 

.0399 

9 

.2385 

.0795 

.0996 

.0332 

10 

.2248 

.0711 

.0891 

.0282 

11 

.2133 

.0643 

.0806 

.0243 

12 

.2034 

.0587 

.0736 

.0212 

13 

.1947 

.0540 

.0677 

.0188 

14 

.1871 

.0500 

.0627 

.0167 

15 

.1803 

.0465 

.0583 

.0151 

16 

.1742 

.0435 

.0546 

.0136 

17 

.1686 

.0409 

.0513 

.0124 

18 

.1636 

.0386 

.0483 

.0114 

19 

.1590 

.0365 

.0457 

.0105 

20 

.1547 

.0346 

.0434 

.0097 

21 

.1508 

.0329 

.0412 

.0090 

22 

.1472 

.0314 

.0393 

.0084 

23 

.1438 

.0300 

.0376 

.0078 

24 

.1406 

.0287 

.0360 

.0073 

25 

.1377 

.0275 

.0345 

.0069 

26 

.1349 

.0265 

.0332 

.0065 

27 

.1323 

.0255 

.0319 

.0061 

28 

.1298 

.0245 

.0307 

.0058 

29 

.1275 

.0237 

.0297 

.0055 

30 

.1252 

.0229 

.0287 

.0052 

31 

.1231 

.0221 

.0277 

.0050 

32 

.1211 

.  .0214 

.0268 

.0047 

33 

.1192 

.0208 

.0260 

.0045 

34 

.1174 

.0201 

.0252 

.0043 

35 

.1157 

.0196 

.0245 

.0041 

36 

.1140 

.0190 

.0238 

.0040 

37 

.1124 

.0185 

.0232 

.0038 

38 

.1109 

.0180 

.0225 

.0037 

39 

.1094 

.0175 

.0220 

.0035 

40 

.1080 

.0171 

.0214 

.0034 

45 

.1017 

.0152 

.0190 

.0028 

50 

.0964 

.0136 

.0171 

.0024 

55 

.0918 

.0124 

.0155 

.0021 

60 

.0878 

.0113 

.0142 

.0018 

65 

.0843 

.0105 

.0131 

.0016 

70 

.0812 

.0097 

.0122 

.0015 

75 

.0784 

.0091 

.0113 

.0013 

80 

.0759 

.0085 

.0106 

.0012 

85 

.0736 

.0080 

.0100 

.0011 

90 

.0715 

.0075 

.0094 

.0010 

95 

.06% 

.0071 

.0089 

.0009 

100 

.0678 

.0068 

.0085 

.0008 

64 


MATHEMATICAL  TABLES 


COMPOUND  INTEREST.     AMOUNT  OF  A  GIVEN  PRINCIPAL 

The  amount  A  at  the  end  of  n  years  of  a  given  principal  P  placed  at  compouni 
interest  to-day  is  A  =  P  X  x  or  A  =  P  X  y  or  A  =  P  X  z,  according  as  the  interes 
(at  the  rate  of  r  per  cent,  per  annum)  is  compounded  annually,  semi-annually,  o 
quarterly;  the  factor  x  or  y  or  z  being  taken  from  the  following  tables. 

Values  of  x.    (Interest  compounded  annually;  A  =  P  X  £.) 


Years 

r  =  2 

2H 

3 

?M 

4 

4$* 

5 

6 

7 

i 

1.0200 

1.0250 

1.0300 

1.0350 

1.0400 

1.0450 

1.0500 

1.0600 

1.0700 

2 

1.0404 

1.0506 

1.0609 

1.0712 

1.0816 

1.0920 

1.1025 

.1236 

1.1449 

3 

1.0612 

1.0769 

1.0927 

1.1087 

1.1249 

1.1412 

1.1576 

.1910 

1.2250 

4 

1.0824 

1.  1038 

1.1255 

1.1475 

1.1699 

1.1925 

12155 

.2625 

1.3108 

3 

5 

1.1041 

1.1314 

1.1593 

1.1877 

1.2167 

1.2462 

1.2763 

.3382 

1.4026 

6 

1.1262 

1.1597 

1.1941 

1.2293 

1.2653 

1.3023 

1.3401 

.4185 

1.5007 

1 

7 

1.1487 

1.1887 

1.2299 

1.2723 

1.3159 

1.3609 

1.4071 

.5036 

1.6058 

** 

8 

1.1717 

1.2184 

1.2668 

1.3168 

1.3686 

1.4221 

1.4775 

.5938 

1.7182 

Ja 

9 

1.1951 

1.2489 

1.3048 

1.3629 

1.4233 

1.4861 

1.5513 

1.6895 

1.8385 

•^  «' 

10 

1.2190 

1.2801 

1.3439 

1.4106 

1.4802 

1.5530 

1.6289 

1.7908 

1.9672 

if 

11 

1.2434 

1.3121 

1.3842 

1.4600 

1.5395 

1.6239 

1.7103 

1.8983 

2.1049 

t,  O 

12 

1.2682 

1.3449 

1.4258 

1.5111 

1.6010 

1.6959 

1.7959 

2.0122 

2.2522 

^"vT 

13 

1.2936 

1.3785 

1.4685 

1.5640 

1.6651 

1.7722 

1.8856 

2.1329 

2.4098 

v*^ 

14 

13195 

1.4130 

1.5126 

1.6187 

1.7317 

1.3519 

1.9799 

2.2609 

2.5785 

a~^~ 

15 

1.3459 

1.4483 

1.5580 

1.6753 

1.8009 

19353 

2.0789 

2.3966 

2.7590 

S^i 

16 

1.3728 

1.4845 

1.6047 

1.7340 

1.8730 

2.0224 

2.1829 

2.5404 

2.9522 

2  !l 

17 

1.4002 

1.5216 

1.6528 

1.7947 

1.9479 

2.1134 

2.2920 

2.6928 

3.1588 

.5  H 

18 

1.4282 

1.5597 

1.7024 

1.8575 

2.0258 

2.2085 

2.4066 

2.8543 

3.3799 

0 

19 

1.4568 

1.5987 

1.7535 

1.9225 

2.1068 

2.3079 

2.5270 

3.0256 

3.6165 

20 

1.4859 

1.6386 

1.8061 

1.9898 

2.1911 

2.4117 

2.6^533 

3.2071 

3.8697 

-2 

25 

1.6406 

1.8539 

2.0938 

2.3632 

2.6658 

3.0054 

3.3864 

4.2919 

5.4274 

.2 

30 

1.8114 

2.0976 

2.4273 

2.8068 

3.2434 

3.7453 

4.3219 

5.7435 

7.6123 

g2 

40 

2.2080 

2.6851 

3.2620 

3.9593 

4.8010 

5.8164 

7.0400 

10.286 

14.974 

50 

2.6916 

3.4371 

4.3839 

5.5849 

7.1067 

9.0326 

1  1  .467 

18.420 

29.457 

60 

3.2810 

4.3998 

5.8916 

7.8781 

10.520 

14.027 

18.679 

32.988 

57.946 

Values  of  y.    (Interest  compounded  semi-annually;  A  =  P  X  y.) 


Years 

r=2 

2H 

3 

3H 

4 

4H 

5 

6 

7 

1 

1.0201 

1.0252 

1.0302 

1.0353 

1.0404 

1  .0455 

1.0506 

1.0609 

1.0712 

2 

1.0406 

1.0509 

1.0614 

1.0719 

1.0824 

1.0931 

1.1038 

1.1255 

1.1475 

3 

1.0615 

1.0774 

1.0934 

1.1097 

1.1262 

1.1428 

1.1597 

1.1941 

1  .2293 

4 

1.0829 

1.1045 

1.1265 

1.1489 

1.1717 

1.1948 

1.2134 

1.2668 

1.3168 

5 

1.1046 

1.1323 

1.1605 

1.1894 

1.2190 

1.2492 

1.2801 

1.3439 

1.4106 

6 

1.1268 

1.1608 

1.1956 

1.2314 

1.2682 

1.3060 

1.3449 

1.4258 

1.5111 

7 

1.1495 

1.1900 

1.2318 

1.2749 

1.3195 

1.3655 

1.4130 

1.5126 

1  .61  87 

«5 

8 

1.1726 

1.2199 

1.2690 

1.3199 

1.3728 

1.4276 

1.4845 

1.6047 

1.7340 

g^ 

9 

1.1961 

1.2506 

1.3073 

1.3665 

1.4282 

1.4926 

1.5597 

1.7024 

1.8575 

g 

10 

1.2202 

1.2820 

1.3469 

1.4148 

1.4859 

1.5605 

1.6386 

1.8061 

1.9898 

~£- 

11 

1.2447 

1.3143 

1.3876 

1.4647 

1.5460 

1.6315 

1.7216 

1.9161 

2.1315 

v~' 

12 

1.2697 

1.3474 

1.4295 

1.5164 

1.6084 

1.7058 

1.8087 

2.0328 

2.2833 

+ 

13 

1.2953 

1.3812 

1.4727 

1.5700 

1.6734 

1.7834 

1.9003 

2.1566 

2.4460 

•H 

14 

1.3213 

1.4160 

1.5172 

1.6254 

1.7410 

1.8645 

1.9965 

2.2879 

2.6202 

II 

15 

1.3478 

1.4516 

1.5631 

1.6828 

1.8114 

1.9494 

2.0976 

2.4273 

2.8068 

a 

16 

1.3749 

1.4881 

1.6103 

1.7422 

1.8845 

2.0381 

2.2038 

2.5751 

3.0067 

17 

1  .4026 

1.5256 

1.6590 

1.8037 

1.9607 

2.1308 

2.3153 

2.7319 

3.2209 

jj 

18 

1.4308 

1.5639 

1.7091 

1.8674 

2.0399 

2.2278 

2.4325 

2.8983 

3.4503 

"9 

19 

1.4595 

1.6033 

1.7608 

1.9333 

2.1223 

2.3292 

2.5557 

3.0748 

3.6960 

S 

20 

1.4889 

1.6436 

1.8140 

2.0016 

2.2080 

2.4352 

2.6851 

3.2620 

3.9593 

fe 

25 

1.6446 

1.8610 

2.1052 

2.3808 

2.6916 

3.0420 

3.4371 

4.3839 

5.5849 

30 

1.8167 

2.1072 

2.4432 

2.8318 

3.2810 

3.8001 

4.3998 

5.8916 

7.8781 

40 

2.2167 

2.7015 

3.2907 

4.0064 

4.8754 

5.9301 

7.2096 

10.641 

15.676 

50 

2.7048 

3.4634 

4.4320 

5.6682 

7.2446 

9.2540 

11.814 

19.219 

31.191 

60 

3.3004 

4.4402 

5.9693 

8.0192 

10.765 

14.441 

19.358 

34.711 

62.064 

MATHEMATICAL  TABLES  65 

Values  of  z.  (Interest  compounded  quarterly;  A  =  P  X  2;  see  opposite  page) 


Years 

•  =  2 

2H 

3 

3^ 

4 

4>i 

5 

6 

7 

1 

.0202 

.0252 

1.0303 

1.0355 

1.0406 

1.0458 

1.0509 

1.0614 

1.0719 

2 

.0407 

.0511 

1.0616 

1.0722 

1.0829 

1.0936 

.1045 

.1265 

1.1489 

3 

.0617 

.0776 

1.0938 

1.1102 

1.1268 

1.1437 

.1608 

.1956 

1.2314 

4 

.0831 

.1048 

1.1270 

1.1496 

1.1726 

1.1960 

.2199 

.2690 

1.3199 

5 

.1049 

.1327 

1.1612 

1.1903 

1.2202 

1.2508 

.2820 

.3469 

1.4148 

6 

.1272 

.1613 

1.1964 

1.2326 

1.2697 

1.3080 

.3474 

.4295 

1.5164 

_« 

7 

.1499 

.1906 

1.2327 

1.2763 

1.3213 

1.3679 

.4160 

.5172 

1.6254 

X 

8 

.1730 

.2206 

1.2701 

1.3215 

1.3749 

1.4305 

.4881 

1.6103 

1.7422 

o" 

9 

.1967 

.2514 

1.3086 

1.3684 

1.4308 

1.4959 

1.5639 

1.7091 

1.8674 

3 

10 

.2208 

.2830 

1.3483 

1.4169 

1.4889 

1.5644 

1.6436 

1.8140 

2.0016 

^ 

11 

.2454 

.3154 

1.3893 

1.4672 

1.5493 

1.6360 

1.7274 

1.9253 

2.1454 

12 

.2705 

.3486 

1.4314 

1.5192 

1.6122 

1.7108 

1.8154 

2.0435 

2.2996 

< 

13 

.2961 

.3826 

1.4748 

1.5731 

1.6777 

1.7891 

1.9078 

2.1689 

2.4648 

;-*. 

14 

.3222 

.4175 

1.5196 

1.6288 

1.7458 

1.8710 

2.0050 

2.3020  • 

2.6420 

II 

15 

.3489 

.4533 

1.5657 

1.6866 

1.8167 

1.9566 

2.1072 

2.4432 

2.8318 

M 

16 

.3760 

.4900 

1.6132 

1.7464 

1.8905 

2.0462 

2.2145 

2.5931 

3.0353 

17 

.4038 

.5276 

1.6621 

1.8083 

1.9672 

2.1398 

2.3274 

2.7523 

3.2534 

J5 

18 

.4320 

.5661 

1.7126 

1.8725 

2.0471 

2.2378 

2.4459 

2.9212 

3.4872 

19 

.4609 

.6056 

,1.7645 

1.9389 

2.1302 

2.3402 

2.5705 

3.1004 

3.7378 

o 

20 

.4903 

.6462 

1.8180 

2.0076 

2.2167 

2.4473 

2.7015 

3.2907 

4.0064 

A 

25 

.6467 

.8646 

2.1111 

2.3898 

2.7048 

3.0609 

3.4634 

4.4320 

5.6682 

30 

.8194  i 

.1121 

2.4514 

2.8446 

3.3004 

3.8285 

4.4402 

5.9693 

8.0192 

40 

!.22ii  ; 

'.7098 

3.3053 

4.0306 

4.9138 

5.9892 

7.2980 

10.828 

16.051 

50 

5.7115  3 

.4768 

4.4567 

5.7110 

7.3160 

9.3693 

11.995 

19.643 

32.128 

60 

L3102  ^ 

L4608 

6.0092 

8.0919 

10.893 

14.657 

19.715 

35.633 

64.307 

AMOUNT  OP  AN  ANNUITY 

The  amount  S  accumulated  at  the  end  of  n  years  by  a  given  annual  payment  Y  set 
aside  at  the  end  of  each  year  is  S  =*  F  X  »,  where  the  factor  v  is  to  be  taken  from  the 
following  table.  (Interest  at  r  per  cent,  per  annum,  compounded  annually.) 

Values  of  v 


Years 

r=  2 

2H 

3 

Hi 

4 

^ 

5 

6 

7 

1 

1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

1  .0000 

2 

2.0200 

2.0250 

2.0300 

2.0350 

2.0400 

.2.0450 

2.0500 

2.0600 

2.0700 

3 

3.0604 

3.0756 

3.0909 

3.1062 

3.1216 

3.1370 

3.1525 

3.1836 

3.2149 

§ 

4 

4.1216 

4.1525 

4.1836 

4.2149 

4.2465 

4.2782 

4.3101 

4.3746 

4.4399 

O 

5 

5.2040 

5.2563 

5.3091 

5.3625 

5.4163 

5.4707 

5.5256 

5.6371 

5.7507 

> 

6 

6.3081 

6.3877 

6.4684 

6.5502 

6.6330 

6.7169 

6.8019 

6.9753 

7.1533 

.1. 

7 

7.4343 

7.5474 

7.6625 

7.7794 

7.8983 

8.0192 

8.1420 

8.3938 

8.6540 

I 

8 

8.5830 

8.7361 

8.8923 

9.0517 

9.2142 

9.3800 

9.5491 

9.8975 

10.260 

^ 

9 

9.7546 

9.9545 

10.159 

10.368 

10.583 

10.802 

11.027 

11.491 

11.978 

1       ; 

10 

10.950 

11.203 

11.464 

11.731 

12.006 

12.288 

12.578 

13.181 

13.816 

«    S 

11 

12.169 

12.483 

12.808 

13.142 

13.486 

13.841 

14.207 

14.972 

15.784 

1—1  o 

12 

13.412 

13.796 

14.192 

14.602 

15.026 

15.464 

15.917 

16.870 

17.888 

!> 

13 

14.680 

15.140 

15.618 

16.113 

16.627 

17.160 

17.713 

18.882 

20.141 

14 

15.974 

16.519 

17.086 

17.677 

18.292 

18.932 

19.599 

21.015 

22.550 

>  •!• 

15 

17.293 

17.932 

18.599 

19.296 

20.024 

20.784 

21.579 

23.276 

25.129 

+  7 

16 

18.639 

19.380 

20.157 

20.971 

21.825 

22.719 

23.657 

25.673 

27.888 

^  1 

17 

20.012 

20.865 

21.762 

22.705 

23.698 

24.742 

25.840 

28.213 

30.840 

"  vS 

18 

21.412 

22.386 

23.414 

24.500 

25.645 

26.855 

28.132 

30.906 

33.999 

^ 

19 

22.841 

23.946 

25.117 

26.357 

27.671 

29.064 

30.539 

33.760 

37.379 

11    0 

20 

24.297 

25.545 

26.870 

28.280 

29.778 

31.371 

33.066 

36.786 

40.995 

25 

32.030 

34.158 

36.459 

38.950 

41.646 

44.565 

47.727 

54.865 

63.249 

a 

30 

40.568 

43.903 

47.575 

51.623 

56.085 

61.007 

66.439 

79.058 

94.461 

1 

40 

60.402 

67.403 

75.401 

84.550 

95.026 

107.03 

120.80 

154.76 

199.64 

i 

50 

84.579 

97.484 

112.80 

131.00 

152.67 

178.50 

209.35 

290.34 

406.53 

PH 

60 

114.05 

135.99 

163.05 

196.52 

237.99 

289.50 

353.58 

533.13 

813.52 

66 


MATHEMATICAL  TABLES 


PRINCIPAL  WHICH  WILL  AMOUNT  TO  A  GIVEN  SUM 

The  principal  P,  which,  if  placed  at  compound  interest  to-day,  will  amount  to  a  giv< 
sum  A  at  the  end  of  n  years  is  P  =  A  X  x'  or  P  =  A  X  y'  or  P  =  A  X  z',  according 
the  interest  (at  the  rate  of  r  per  cent,  per  annum)  is  compounded  annually,  semi-annuall 
or  quarterly:  the  factor  x'  or  y'  or  z'  being  taken  from  the  following  tables. 
Values  of  x'.     (Interest  compounded  annually;  P  =  A  X  x') 


Years 

r  =  2 

2H 

3 

sw 

4 

4H 

5 

6 

7 

1 

.98039 

.97561 

.97087 

.96618 

.96154 

.95694 

.95238 

.94340 

.93458 

2 

.96117 

.95181 

.94260 

.93351 

.92456 

.91573 

.90703 

.89000 

.87344 

3 

.94232 

.92860 

.91514 

.90194 

.88900 

.87630 

.86384 

.83962 

.81630 

4 

.92385 

.90595 

.88849 

.87144 

.85480 

.83856 

.82270 

.79209 

.76290 

. 

5 

.90573 

C88385 

.86261 

.84197 

.82193 

.80245 

.78353 

.74726 

.71299 

i-H 

6 

.88797 

.86230 

.83748 

.81350 

.79031 

.76790 

.74622 

.70496 

.66634 

7 

.87056 

.84127 

.81309 

.78599 

.75992 

.73483 

.71068 

.66506 

.62275 

8 

.85349 

.82075 

.78941 

.75941 

.73069 

.70319 

.67684 

.62741 

.58201 

1! 
1  ^ 

9 

.83676 

.80073 

.76642 

.73373 

.70259 

.67290 

.64461 

.59190 

.54393 

10 

.82035 

.78120 

.74409 

.70892 

.67556 

.64393 

.61391 

.55839 

.50835 

I 

11 

.80426 

.76214 

.72242 

.68495 

.64958 

.61620 

.58468 

.52679 

.47509 

3 

12 

.78849 

.74356 

.70138 

.66178 

.62460 

.58966 

.55684 

.49697 

.44401 

13 

.77303 

.72542 

.68095 

.63940 

.60057 

.56427 

.53032 

.46884 

.41496 

4- 

14 

.75788 

.70773 

.66112 

.61778 

.57748 

.53997 

.50507 

.44230 

.38783 

i-H 

15 

.74301 

.69047 

.64186 

.59689 

.55526 

.51672 

.48102 

.41727 

.36245 

D 

16 

.72845 

.67362 

.62317 

.57671 

.53391 

.49447 

.4581  1 

.39365 

.33873 

17 

.71416 

.65720 

.60502 

.55720 

.51337 

.47318 

.43630 

.37136 

.31657 

"« 

18 

.70016 

.64117 

.58739 

.53836 

.49363 

.45280 

.41552 

.35034 

.29586 

cj 

19 

.68643 

.62553 

.57029 

.52016 

.47464 

.43330 

.39573 

.33051 

.27651 

3 

20 

.67297 

.61027 

.55368 

.50257 

.45639 

.41464 

.37689 

.31180 

.25842 

jjj 

25 

.60953 

.53939 

.47761 

.42315 

.37512 

.33273 

.29530 

.23300 

.18425 

o 

30 

.55207 

.47674 

.41199 

.35628 

.30832 

.26700 

.23138 

.17411 

.13137 

n 

40 

.45289 

.37243 

.30656 

.25257 

.20829 

.17193 

.14205 

.09722 

.06678 

50 

.37153 

.29094 

.2281  1 

.17905 

.14071 

.11071 

.08720 

.05429 

.03395 

60 

.30478 

.22728 

.16973 

.12693 

.09506 

.07129 

.05354 

.03031 

.01726 

Values  of  y'.     (Interest  compounded  semi-annually;  P  =  A  X  y') 


Years 

r  =  2 

2V 

3 

W 

4 

4J/2 

5 

6 

7 

1 

.98030 

.97546 

.97066 

.96590 

.96117 

.95647 

.95181 

.94260 

.93351 

2 

.96098 

.95152 

.94218 

.93296 

.92385 

.91484 

.90595 

.88849 

.87144 

3 

.94205 

.92817 

.91454 

.90114 

.88797 

.87502 

.86230 

.83748 

.81350 

4 

.92348 

.90540 

.88771 

.87041 

.85349 

.83694 

.82075 

.78941 

.75941 

.' 

5 

.90529 

.88318 

.86167 

.84073 

.82035 

.80051 

.78120 

.74409 

.70892 

\ 

6 

.88745 

.86151 

.83639 

.81206 

.78849 

.76567 

.74356 

.70138 

.66178 

7 

.86996 

.84037 

.81185 

.78436 

.75788 

.73234 

.70773 

.66112 

.61778 

8 

.85282 

.81975 

.78803 

.75762 

.72845 

.70047 

.67362 

.62317 

.57671 

* 

9 

.83602 

.79963 

.76491 

.73178 

.70016 

.66998 

.64117 

.58739 

.53836 

~Z 

10 

.81954 

.78001 

.74247 

.70682 

.67297 

.64082 

.61027 

.55368 

.50257 

§ 

11 

.80340 

.76087 

.72069 

.68272 

.64684 

.61292 

.58086 

.52189 

.46915 

^ 

12 

.78757 

.74220 

.69954 

.65944 

.62172 

.58625 

.55288 

.49193 

.43796 

£, 

13 

.77205 

.72398 

.67902 

.63695 

.59758 

.56073 

.52623 

.46369 

.40884 

4. 

14 

.75684 

.70622 

.65910 

.61523 

.57437 

.53632 

.50088 

.43708 

.38165 

i 

15 

.74192 

.68889 

.63976 

.59425 

.55207 

.51298 

.47674 

.41199 

.35628 

*-* 

16 

.72730 

.67198 

.62099 

.57398 

.53063 

.49065 

.45377 

.38834 

.33259 

n 

17 

.71297 

.65549 

.60277 

.55441 

.51003 

.46930 

.43191 

.36604 

.31048 

*a» 

18 

.69892 

.63941 

.58509 

.53550 

.49022 

.44887 

.41109 

.34503 

.28983 

19 

.68515 

.62372 

.56792 

.51724 

.47119 

.42933 

.39128 

.32523 

.27056 

J2 

20 

.67165 

.60841 

.55126 

.49960 

.45289 

.41065 

.37243 

.30656 

.25257 

s 

25 

.60804 

.53734 

.47500 

.42003 

.37153 

.32873 

.29094 

.22811 

.17905 

30 

.55045 

.47457 

.40930 

.35313 

.30478 

.26315 

.22728 

.16973 

.12693 

£ 

40 

.45112 

.37017 

.30389 

.24960 

.20511 

.16863 

.13870 

.09398 

.06379 

50 

.36971 

.28873 

.22563 

.17642 

.13803 

.10806 

.08465 

.05203 

.03206 

60 

.30299 

.22521 

.16752 

.12470 

.09289 

.06925 

.05166 

.02881 

.01611 

MATHEMATICAL  TABLES 


67 


Values  of  «'.     (Interest  compounded  quarterly;  P=>A  X  z'',  see  opposite  page) 


Years|  r  =  2 

2tt 

3 

JH 

4 

4J4 

5 

6 

7 

1 

.98025 

.97539 

.97055 

.96575 

.96098 

.95624 

.95152 

.94218 

.93296 

2 

.96089 

.95138 

.94198 

.93268 

.92348 

.91439 

.90540 

.88771 

.87041 

3 

.94191 

.92796 

.91424 

.90074 

.88745 

.87437 

.86151 

.83639 

.81206 

4 

.92330 

.90512 

.88732 

.86989 

.85282 

.8361  1 

.81975 

.78803 

.75762 

5 

.90506 

.88284 

.86119 

.84010 

.81954 

.79952 

.78001 

.74247 

.70682 

\ 
1-1 

6 

.88719 

.86111 

.83583 

.81132 

.78757 

.76453 

.74220 

.69954 

.65944 

* 

7 

.86966 

.83991 

.81122 

.78354 

.75684 

.73107 

.70622 

.65910 

.61523 

8 

.85248 

.81924 

.78733 

.75670 

.72730 

.69908 

.67198 

.62099 

.57390 

j 

9 

.83564 

.79908 

.76415 

.73079 

.69892 

.66849 

.63941 

.58509 

.53550 

JL* 

10 

.81914 

.77941 

.74165 

.70576 

.67165 

.63923 

.60841 

.55126 

.49960 

o 

11 

.80296 

.76022 

.71981 

.68159 

.64545 

.61126 

.57892 

.51939 

.46611 

V 

12 

.78710 

.74151 

.69861 

.65825 

.62026 

.58451 

.55086 

.48936 

.43486 

^ 

13 

.77155 

.72326 

.67804 

.63570 

.59606 

.55893 

.52415 

.46107 

.40570 

-j- 

14 

.75631 

.70546 

.65808 

.61393 

.57280 

.53447 

.49874 

.43441 

.37851 

15 

.74137 

.68809 

.63870 

.59291 

.55045 

.51108 

.47457 

.40930 

.35313 

'•— 

16 

.72673 

.67115 

.61989 

.57260 

.52897 

.48871 

.45156 

.38563 

.32946 

II 

17 

.71237 

.65464 

.60164 

.55299 

.50833 

.46733 

.42967 

.36334 

.30737 

%* 

18 

.69830 

.63852 

.58392 

.53405 

.48850 

.44687 

.40884 

.34233 

.28676 

19 

.68451 

.62281 

.56673 

.51576 

.46944 

.42732 

.38903 

.32254 

.26754 

Js 

"3 

20 

.67099 

.60748 

.55004 

.49810 

.45112 

.40862 

.37017 

.30389 

.24960 

25 

.60729 

.53630 

.47369 

.41845 

.36971 

.32670 

.28873 

.22563 

.17642 

o 

30 

.54963 

.47347 

.40794 

.35154 

.30299 

.26120 

.22521 

.16752 

.12470 

£ 

40 

.45023 

.36903 

.30255 

.24810 

.20351 

.16697 

.13702 

.09235 

.06230 

50 

.36880 

.28762 

.22438 

.17510 

.13669 

.10673 

.08337 

.05091 

.03113 

60 

.30210 

.22417 

.16641 

.12358 

.09181 

.06823 

.05072 

.02806 

.01555 

ANNUITY   WHICH   WILL  AMOUNT  TO  A  GIVEN  SUM  (SINKING 
FUND) 

The  annual  payment,  Y,  which,  if  set  aside  at  the  end  of  each  year,  will  amount  with 
accumulated  interest  to  a  given  sum  S  at  the  end  of  n  years  is  Y  =  S  X  v',  where  the 
factor  v'  is  given  below.  (Interest  at  r  per  cent,  per  annum,  compounded  annually.) 

Values  of  »' 


Years 

r=  2 

2H 

3 

3H 

4 

^ 

5 

6 

7 

2 

.49505 

.49383 

.49261 

.49140 

.49020 

.48900 

.48780 

.48544 

.48309 

»' 

3 

.32675 

.32514 

.32353 

.32193 

.32035 

.31877 

.31721 

.31411 

.31105 

^^ 

4 

.24262 

.24082 

.23903 

.23725 

.23549 

.23374 

.23201 

.22859 

.22523 

P 

5 

.19216 

.19025 

.18835 

.18648 

.18463 

.18279 

.18097 

.17740 

.17389 

„ 

6 

.15853 

.15655 

.15460 

.15267 

.15076 

.14888 

.14702 

.14336 

.13980 

'"H 

7 

.13451 

.13250 

.13051 

.12854 

.12661 

.12470 

.12282 

.11914 

.11555 

1 

8 

.11651 

-.11447 

.11246 

.11048 

.10853 

.10661 

.10472 

.10104 

.09747 

4, 

9 

.10252 

.10046 

.09843 

.09645 

.09449 

.09257 

.09069 

.08702 

.08349 

§ 

10 

.09133 

.08926 

.08723 

.08524 

.08329 

.08138 

.07950 

.07587 

.07238 

11 

.08218 

.0801  1 

.07808 

.07609 

.07415 

.07225 

.07039 

.06679 

.06336 

I* 

12 

.07456 

.07249 

.07046 

.06848 

.06655 

.06467 

.06283 

.05928 

.05590 

13 

.06812 

.06605 

.06403 

.06206 

.06014 

.05828 

.05646 

.05296 

.04965 

14 

.06260 

.06054 

.05853 

.05657 

.05467 

.05282 

.05102 

.04758 

.04434 

j* 

15 

.05783 

.05577 

.05377 

.05183 

.04994 

.0481  1 

.04634 

.04296 

.03979 

•!• 

16 

.05365 

.05160 

.04961 

.04768 

.04582 

.04402 

.04227 

.03895 

.03586 

17 

.04997 

.04793 

.04595 

.04404 

.04220 

.04042 

.03870 

.03544 

.03243 

§ 

18 

.04670 

.04467 

.04271 

.04082 

.03899 

.03724 

.03555 

.03236 

.02941 

19 

.04378 

.04176 

.03981 

.03794 

.03614 

.03441 

.03275 

.02962 

.02675 

| 

20 

.04116 

.03915 

.03722 

.03536 

.03358 

.03188 

.03024 

.02718 

.02439 

..  ^ 

25 

.03122 

.02928 

.02743 

.02567 

.02401 

.02244 

.02095 

.01823 

.01581 

J2  " 

30 

.02465 

.02278 

.02102 

.01937 

.01783 

.01639 

.01505 

.01265 

.01059 

l> 

40 

.01656 

.01484 

.01326 

.01183 

.01052 

.00934 

.00828 

.00646 

.00467 

|H 

0 

50 

.01182 

.01026 

.00887 

.00763 

.00655 

.00560 

.00478 

.00344 

.00238 

fe 

60    .00877 

.00735 

.00613 

.00509 

.00420 

.00345 

.00283 

.00188 

.00121 

68 


MATHEMATICAL  TABLES 


PRESENT  WORTH  OF  AN  ANNUITY 

The  capital  C,  which,  if  placed  at  interest  to-day,  will  provide  for  a  given  annual 
payment  Y  for  a  term  of  n  years  before  it  is  exhausted  is  C  =  Y  X  w,  where  the  factor 
w  is  given  below.  (Interest  at  r  per  cent,  per  annum,  compounded  annually.) 

Values  of  w 


Years|  r  =2 

2H 

3 

3M 

4 

4H 

5 

6 

7 

1 

0.9804 

0.9756 

0.9709 

0.9662 

0.9615 

0.9569 

0.9524 

0.9434 

0.9346 

2 

1.9416 

1.9274 

1.9135 

1.8997 

1.8861 

1.8727 

1.8594 

1.8334 

1.8080 

N 

3 

2.8839 

2.8560 

2.8286 

2.8016 

2.7751 

2.7490 

2.7232 

2.6730 

2.6243 

\ 

4 

3.8077 

3.7620 

3.7171 

3.6731 

3.6299 

3.5875 

3.5460 

3.4651 

3.3872 

1 

5 

4.7135 

4.6458 

4.5797 

4.5151 

4.4518 

4.3900 

4.3295 

4.2124 

4.1002 

§"" 

6 

5.6014 

5.5081 

5.4172 

5.3286 

5.2421 

5.1579 

5.0757 

4.9173 

4.7665 

7 

6.4720 

6.3494 

6.2303 

6.1145 

6.0021 

5.8927 

5.7864 

5.5824 

5.3893 

\ 

8 

7.3255 

7.1701 

7.0197 

6.8740 

6.7327 

6.5959 

6.4632 

6.2098 

5.9713 

*• 

9 

8.1622 

7.9709 

7.7861 

7.6077 

7.4353 

7.2688 

7.1078 

6.8017 

6.5152 

•1- 

10 

8.9826 

8.7521 

8.5302 

8.3166 

8.1109 

7.9127 

7.7217 

7.3601 

7.0236 

11 

9.7868 

9.5142 

9.2526 

9.0016 

8.7605 

8.5289 

8.3064 

7.8869 

7.4987 

i 

12 

10.575 

10.258 

9.9540 

9.6633 

9.3851 

9.1186 

8.8633 

8.3838 

7.9427 

13 

11.348 

10.983 

10.635 

10.303 

9.9856 

9.6829 

9.3936 

8.8527 

8.3577 

I 

14 

12.106 

11.691 

11.296 

10.921 

10.563 

10.223 

9.8986 

9.2950 

8.7455 

\ 

15 

12.849 

12.381 

11.938 

11.517 

11.118 

10.740 

10.380 

9.7122 

9.1079 

^ 

16 

13.578 

13.055 

12.561 

12.094 

11.652 

11.234 

10.838 

10.106 

9.4466 

+ 

17 

1  4.292 

13.712 

13.166 

12.651 

12.166 

11.707 

11.274 

10.477 

9.7632 

18 

14.992 

14.353 

13.754 

13.190 

12.659 

12.160 

11.690 

10.828 

10.059 

Zl 

19 

15.678 

14.979 

14.324 

13.710 

13.134 

12.593 

12.085 

11.15.8 

10.336 

1 

20 

16.351 

15.589 

14.877 

14.212 

13.590 

13.008 

12.462 

11.470 

10.594 

a  ^ 

25 

19.523 

18.424 

17.413 

16.482 

15.622 

14.828 

14.094 

12.783 

11.654 

3  T 

30 

22.396 

20.930 

19.600 

18.392 

17.292 

16.289 

15.372 

13.765 

12.409 

s  " 

40 

27.355 

25.103 

23.115 

21.355 

19.793 

18.402 

17.159 

15.046 

13.332 

o  5 

50 

31.424 

28.362 

25.730 

23.456 

21.482 

19.762 

18.256 

15.762 

13.801 

HH 

60 

34.761 

30.909 

27.676 

24.945 

22.623 

20.638 

18.929 

16.161 

14.039 

ANNUITY  PROVIDED  FOR  BY  A  GIVEN  CAPITAL 

The  annual  payment  Y  provided  for  for  a  term  of  n  years  by  a  given  capital  C  placed 
at  interest  to-day  is  Y  =  C  X  w'  .  (Interest  at  r  per  cent,  per  annum,  compounded 
annually;  the  fund  supposed  to  be  exhausted  at  the  end  of  the  term.) 

Values  of  w' 


Years 

r  =  2 

2% 

3 

3H 

4 

4H 

5 

6 

7 

2 

.51505 

.51883 

.52261 

.52640 

.53020 

.53400 

.53780 

.54544 

.55309 

3 

.34675 

.35014 

.35353 

.35693 

.36035 

.36377 

.36721 

.37411 

.38105 

• 

4 

.26262 

.26582 

.26903 

.27225 

.27549 

.27874 

.28201 

.28859 

.29523 

c; 

5 

.21216 

.21525 

.21835 

.22148 

.22463 

.22779 

.23097 

.23740 

.24389 

§ 

6 

.17853 

.18155 

.18460 

.18767 

.19076 

.19388 

.19702 

.20336 

.20980 

C^ 

7 

.15451 

.15750 

.16051 

.16354 

.16661 

.16970 

.17282 

.17914 

.18555 

^ 

8 

.13651 

.13947 

.14246 

.14548 

.14853 

.15161 

.15472 

.16104 

.16747 

_i_  X. 

9 

.12252 

.12546 

.12843 

.13145 

.13449 

.13757 

.14069 

.14702 

.15349 

^  § 

10 

.11133 

.11426 

.11723 

.12024 

.12329 

.12638 

.12950 

.13587 

.14238 

_  TH 

11 

.10218 

.10511 

.10808 

.11109 

.11415 

.11725 

.12039 

.12679 

.13336 

1    xb 

12 

.09456 

.09749 

.10046 

.10348 

.10655 

.10967 

.11283 

.11928 

.12590 

~  + 

13 

.08812 

.09105 

.09403 

.09706 

.10014 

.10328 

.10646 

.11296 

.11965 

14 

.08260 

.08554 

.08853 

.09157 

.09467 

.09782 

.10102 

.10758 

.11434 

•1'  ve 

15 

.07783 

.08077 

.08377 

.08683 

.08994 

.09311 

.09634 

.10296 

.10979 

Ii 

16 

.07365 

.07660 

.07961 

.08268 

.08582 

.08902 

.09227 

.09895 

.10586 

17 

.06997 

.07293 

.07595 

.07904 

.08220 

.08542 

.08870 

.09544 

.10243 

^T  ^ 

18 

.06670 

.06967 

.07271 

.07582 

.07899 

.08224 

.08555 

.09236 

.09941 

II  II 

19 

.06378 

.06676 

.06981 

.07294 

.07614 

.07941 

.08275 

.08962 

.09675 

20 

.06116 

.06415 

.06722 

.07036 

.07358 

.07688 

.08024 

.08718 

.09439 

9 

25 

.05122 

.05428 

.05743 

.06067 

.06401 

.06744 

.07095 

.07823 

.08581 

•  • 

30 

.04465 

.04778 

.05102 

.05437 

.05783 

.06139 

.06505 

.07265 

.08059 

"5 

40 

.03656 

.03984 

.04326 

.04683 

.05052 

.05434 

.05828 

.06646 

.07467 

g 

50 

.03182 

.03526 

.03887 

.04263 

.04655 

.05060 

.05478 

.06344 

.07238 

0 

60 

.02877 

.03235 

.03613 

.04009 

.04420 

.04845 

.05283 

.06188 

.07121 

fc 

MATHEMATICAL  TABLES 


DECIMAL  EQUIVALENTS 


From   minutes  and 

From  decimal  parts  of 

Common  fractions 

seconds  into  deci- 
mal parts  of  a 
degree 

a  degree  into  minutes 
and   seconds    (exact 
values) 

8       16        32      64 

ths     ths      nds    ths 

Exact 
decimal 
values 

0' 

o°.oooo 

0" 

o°.oooo 

o°.oo 

0' 

0°.50 

30' 

1 

.01  5625 

1 

.0167 

1 

.0003 

1 

0'  36" 

1 

30'  36" 

1         2 

.03  125 

2 

.0333 

2 

.0006 

2 

1'   12" 

2 

31'    12" 

3 

.04  6875 

3 

.05 

3 

.0008 

3 

V  48" 

3 

31'  48" 

1         2         4 

.06  25 

4 

.0667 

4 

.0011 

4 

2'  24" 

4 

32'  24" 

5 

.07  8125 

5' 

.0833 

5" 

.0014 

0°.05 

3' 

0°.55 

33' 

3         6 

.09  375 

6 

.10 

6 

.0017 

6 

y  36" 

6 

33'  36" 

7 

.10  9375 

7 

.1167 

7 

.0019 

7 

4'  12" 

7 

34'    12" 

1248 

.12  5 

8 

.1333 

8 

.0022 

8 

4'  48" 

8 

34'  48" 

9 

.14  0625 

9 

.15 

9 

.0025 

9 

5'  24" 

9 

35'  24" 

5       10 

.15  625 

10' 

0°.1667 

10" 

0°.0028 

0°.10 

6' 

0°.60 

36' 

11 

.17  1875 

1 

.1833 

] 

.0031 

1 

6'  36" 

1 

36'  36" 

3         6       12 

.18  75 

2 

.20 

2 

.0033 

2 

7'   12" 

2 

37'   12" 

13 

.20  3125 

3 

.2167 

3 

.0036 

3 

7'  48" 

3 

37'  48" 

7        14 

.21  875 

4 

.2333 

4 

.0039 

4 

8'  24" 

4 

38'  24" 

15 

.23  4375 

15' 

.25 

15" 

.0042 

0°.15 

9' 

0°.65 

39' 

2         4         8       16 

.25 

6 

.2667 

6 

.0044 

6 

9'  36" 

6 

39'  36" 

17 

.26  5625 

7 

.2833 

7 

.0047 

7 

10'   12" 

7 

40'   12" 

9        18 

.28  125 

8 

.30 

8 

.005 

8 

10'  48" 

8 

40'  48" 

19 

.29  6875 

9 

.3167 

9 

.0053 

9 

11'  24" 

9 

41'   24" 

5        10       20 

.31  25 

20' 

0°.3333 

20" 

0°.0056 

0°.20 

12' 

0°.70 

42' 

21 

.32  8125 

1 

.35 

1 

.0058 

12'  36" 

42'  36" 

11        22 

.34  375 

2 

.3667 

2 

.0061 

2 

13'   12" 

2 

43'   12" 

23 

.35  9375 

3 

.3833 

3 

.0064 

3 

13'  48" 

3 

43'  48" 

3         6       12       24 

.37  5 

4 

.40 

4 

.0067 

4 

14'  24" 

4 

44'  24" 

25 

.39  0625 

25' 

.4167 

25" 

.0069 

0°.25 

15' 

0°.75 

45' 

13       26 

.40  625 

6 

.4333 

6 

.0072 

6 

15'  36" 

6 

45'  36" 

27 

.42  1875 

7 

.45 

7 

.0075 

7 

16'   12" 

7 

46'   12" 

7        14       28 

.43  75 

8 

.4667 

8 

.0078 

8 

16'  48" 

8 

46'  48" 

29 

.45  3125 

9 

.4833 

9 

.0081 

9 

17'  24" 

9 

47'  24" 

15       30 

.46  875 

30' 

0°.50 

30" 

0°.0083 

0°.30 

18' 

0°.80 

48' 

31 

.48  4375 

1 

.5167 

1 

.0086 

1 

18'  36" 

1 

48'  36" 

4         8        16       32 

.50 

2 

.5333 

2 

.0089 

2 

19'   12" 

2 

49'   12" 

33 

.51  5625 

3 

.55 

3 

.0092 

3 

19'  48" 

3 

49'   48" 

17       34 

.53  125 

4 

.5667 

4 

.0094 

4 

20'  24" 

4 

50'  24" 

35 

.54  6875 

35' 

.5833 

35" 

.0097 

0°.35 

21' 

0°.85 

51' 

9        18       36 

.56  25 

6 

.60 

6 

.01 

6 

21'  36" 

6 

51'  36" 

37 

.57  8125 

7 

.6167 

7 

.0103 

7 

22'   12" 

7 

52'   12" 

19       38 

.59  375 

8 

.6333 

8 

.0106 

8 

22'  48" 

8 

52'  48" 

39 

.60  9375 

9 

.65 

9 

.0108 

9 

23'  24" 

9 

53'  24" 

5        10       20       40 

.62  5 

40' 

0°.6667 

40" 

0°.01  1  1 

0°.40 

24' 

0°.90 

54' 

41 

.64  0625 

1 

.6833 

1 

.0114 

1 

24'  36" 

54'  36" 

21        42 

.65  625 

2 

.70 

2 

.0117 

2 

25'   12" 

2 

55'   12" 

43 

.67  1875 

3 

.7167 

3 

.0119 

3 

25'  48" 

3 

55'  48" 

11        22       44 

.68  75 

4 

.7333 

4 

.0122 

4 

26'  24" 

4 

56'  24" 

45 

.70  3125 

45' 

.75 

45" 

.0125 

0°45 

27' 

0°.95 

57' 

23       46 

.71  875 

6 

.7667 

6 

.0128 

6 

27'  36" 

6 

57'  36" 

47 

.73  4375 

7 

.7833 

7 

.0131 

7 

28'   12" 

7 

58'   12" 

6        12       24       48 

.75 

8 

.80 

8 

.0133 

8 

28'  48" 

8 

58'  48" 

49 

.76  5625 

9 

.8167 

9 

.0136 

9 

29'   24" 

9 

59'  24" 

25       50 

.78  125 

50' 

0°.8333 

50" 

0°.0139 

0°.50 

30' 

1°.00 

60' 

51 

.79  6875 

1 

.85 
.8667 

1 

2 

.0142 
.0144 

o°.ooo 

0".0 

13       26       52 
53 

.81  25 
.82  8125 

3 

.8833 

3 

.0147 

1 

3".6 

27       54 

.84  375 

4 

.90 

4 

.015 

2 

7".2 

55 

.85  9375 

55' 

.9167 

55" 

.0153 

3 

10".8 

7        14       28       56 

.87  5 

6 

.9333 

6 

.0156 

4 

14".4 

57 

.89  0625 

7 

.95 

7 

.0158 

0°.005 

18" 

29       58 

.90  625 

8 

.9667 

8 

.0161 

6 

21".6 

59 

.92  1875 

9 

.9833 

9 

.0164 

7 

25".2 

15        30       60 

.93  75 

60' 

1.00 

60" 

0°.0167 

8 

28".8 

61 

.95  3125 

9 

32"  .4 

31        62 

.96  875 

0°.010 

36" 

63 

.98  4375 

WEIGHTS  AND  MEASURES 

BY 
LOUIS  A.  FISCHER 


In  the  United  States  the  measures  of  weight  and  length  commonly  employed 
are  identical  with  the  corresponding  English  units,  but  the  capacity  measures 
differ  from  those  now  in  use  in  the  British  Empire,  the  U.  S.  gallon  being 
defined  as  231  cu.  in.  and  the  bushel  as  2150.42  cu.  in.,  whereas  the  corre- 
sponding British  imperial  units  are,  respectively,  277.418  cu.  in.,  and  2219.344 
cu.  in.  (1  imp.  gal.  =  1.2  U.  S.  gal.,  approx.;  1  imp.  bu.  =  1.03  U.  S.  bu., 
approx.). 

The  metric  system  of  weights  and  measures  was  legalized  and  its  use  made 
permissive  in  the  United  States  by  an  Act  of  Congress,  passed  in  1866. 
In  1872,  by  the  concurrent  action  of  the  principal  governments  of  the  world,  it 
was  agreed  to  establish  an  International  Bureau  of  Weights  and  Measures 
near  Paris. 

Prior  to  1891  the  British  imperial  yard  was  regarded  as  the  real  standard 
of  the  United  States.  In  1891,  the  Office  of  Weights  and  Measures  (now 
Bureau  of  Standards)  fixed  the  value  of  the  United  States  yard  in  terms  of 
the  international  meter,  according  ito  the  ratio:  one  yard  =  3600/3937  meters. 
At  the  same  time,  the  pound  was  fixed  in  terms  of  the  international  kilo- 
gram, according  to  the  relation:  one  pound  =  453.59243  grams. 

U.  S.  Customary  Weights  and  Measures 


Measures  of  Length 


Measures  of  Area 


12  inches 
3  feet 
5H  yards  =  16^  feet 


40  poles  =  220  yards 

8  furlongs  =  1760  yards 

=  5280  feet 

3  miles 

4  inches 

9  inches 


=  1  foot 
=  1  yard 
=  1  rod, 

pole  or 

perch 
=  1  furlong 

=  1  mile 

=  1  league 
=  1  hand 
=  1  span 


=  1  acre 


640  acres  =  1  square  mile  = 


Nautical  Units 
6080.2  feet  =  1  nautical  mile 

6  feet  =  1  fathom 

120  fathoms  =  1  cable  length 

1  nautical  mile  per  hr.  =  1   knot 


Surveyor's  or  Gunter's  Measure 

7.92  inches  =  1  link 

100  links  =  66  ft.  =  4  rods  =  1  chain 
80  chains  =  1  mile 

33H  inches  =  1  vara  (Texas) 


144     square  inches  =  1  square  foot 
9     square  feet      =  1  square  yard 
30^4  square  yards  =  1  square  rod,  pole  or 

perch 
160  square  rods 

=  10  square  chains 

=  43,560sq.  ft, 

=  5645  sq.  varas  (Texas) 

1     ''section" 
of  U.  S.  Govt. 
surveyed 
land 

1  circular  inch 
=  area  of  circle  1  inch 

in  diameter 
1  square  inch      =  1.2732  cir.  in. 
1  circular  mil      =area  of  circle  0.001  in. 

in  diam. 
1,000,000  cir.  mils  =  l  cir.  in. 

Measures  of  Volume 

1728  cubic  inches  =      1  cubic  foot 

27  cubic  feet  =      1  cubic  yard 

1  cord  of  wood          =128  cu.  ft. 
1  perch  of  masonry  =  16K>  to  25  cu.  ft. 


=  0.7854  sq.  in. 


70 


U.  S.    WEIGHTS  AND  MEASURES 


71 


U.  S.  Customary  Weights  and  Measures — (continued) 


Measures  of  Volume 


Weights 

(The  grain  is  the  same  in  all  systems) 


Liquid  or  Fluid  Measure 

4  gills  =  1  pint 

2  pints  =  1  quart 

4  quarts  <=  1  gallon 

7.4805  gallons  =  1  cubic  foot 

(There  is  no  standard  liquid  "barrel.") 

Apothecaries'  Liquid  Measure 

60  minims  =  1  liquid  dram  or  drachm 

8  drams    =  1  liquid  ounce 
16  ounces  =  1  pint 

Water  Measure 

The  Miner's  Inch  is  the  quantity  of 
water  that  will  pass  through  an  orifice  1 
sq.  in.  in  cross-section  under  a  head  of  from 
4  to  6^  in.,  as  fixed  by  statutes,  and  varies 
from  Ho  cu.  ft.  to  Ho  cu.  ft.  per  sec.  The 
units  now  most  in  use  are  1  cu.  ft.  per  sec. 
and  1  gal.  per  sec.,  the  U.  S.  Reclamation 
Service  employing  the  former.  See  p.  260. 

Dry  Measure 

2  pints  =  1  quart 
8  quarts  =  1  peck 
4  pecks  =  1  bushel 

Shipping  Measure 
1  Register  ton  =  100  cu.  ft. 

1  U.  S.  shipping  ton     =   40  cu.  ft. 

=  f 32.14  U.  S.  bu. 
m  131.14  imp.  bu. 
1  British  shipping  ton  =   42  cu.  ft. 

/  32.70  imp.  bu. 
~  \  33.75  U.  S.  bu. 

Board  Measure 

(  144    cu.   in.  =  volume    of 
1  board  foot  =  {  board  1  ft.  sq.  and  1-  in. 

i  thick. 

No.  of  board  feet  in  a  log  =  [\i(d  -  4)pZ,, 
where  d  =  diam.  of  log  (usually  taken  in- 
side the  bark  at  small  end),  in.,  and  L  = 
length  of  log,  ft.  The  4  in.  deducted  are 
an  allowance  for  slab.  This  rule  is  vari- 
ously known  as  the  Doyle,  Conn.  River, 
St.  Croix,  Thurber,  Moore  and  Beeman, 
and  the  Scribner  rule. 


Avoirdupois  Weight 
16  drams  =  437.5  grains  =  1  ounce 
16  ounces  =  7000  grains  =  1  pound 
100  pounds  <=  1  cental 

2000  pounds  «=»  1  short  ton 

2240  pounds  *=  1  long  ton 

Also  (in  Great  Britain): 
14  pounds 
2  stone  =  28  Ib. 
4  quarters  =  1 12  Ib. 


20  hundredweight 


=  1  stone 
=  1  quarter 
=  1  hundred- 
weight (cwt.) 
=  1  long  ton 


24  grains 


Troy  Weight 

=  1  penny- 
weight (dwt.) 
20  pennyweights  =  480  grains  =  1  ounce 
12  ounces  =  5760  grains  =1  pound 

1  Assay  Ton  =  29,167  milligrams,  or 
as  many  milligrams  as  there  are  troy 
ounces  in  a  ton  of  2000  Ib.  avoirdupois. 
Consequently,  the  number  of  milligrams 
of  precious  metal  yielded  by  an  assay  ton 
of  ore  gives  directly  the  number  of  troy 
ounces  that  would  be  obtained  from  a  ton 
of  2000  Ib.  avoirdupois. 

Apothecaries'  Weight 

20  grains  =  1  scruple  3 

3    scruples  =  60    grains  =  1  dram  3 

8  drams  =  1  ounce  5 

12  ounces  =  5760  grains  =  1  pound 

Weight  for  Precious  Stones 

1  carat  =  200  milligrams 

(Adopted    by    practically    all    important 

nations.) 

Circular  Measure 

60  seconds  =  1  minute 
60  minutes  =  1  degree 
90  degrees  =  1  quadrant 
360  degrees  =  circumference 
57.2957795  degrees  =1  radian  (or    angle 
( =  57°  17'44.806")       having  arc  of  length 
equal  to  radius) 


METRIC  SYSTEM 

The  fundamental  unit  of  the  metric  system  is  the  meter — the  unit  of  length, 
from  which  the  units  of  volume  (liter)  and  of  mass  (gram)  are  derived.  All 
other  units  are  the  decimal  subdivisions  or  multiples  of  these.  These  three 
units  are  simply  related :  one  cubic  decimeter  equals  one  liter,  and  one  liter  of 
water  weighs  one  kilogram.  The  metric  tables  are  formed  by  combining 
the  words  "meter,"  "gram,"  and  "liter"  with  numerical  prefixes. 


72 


WEIGHTS  AND  MEASURES 


All  lengths,  areas,  and  cubic  measures  in  the  following  conversion  tables 
are  derived  from  the  international  meter.  The  customary  weights  are  like- 
wise derived  from  the  kilogram.  All  capacities  are  based  on  the  practical 
equivalent:  1  cubic  decimeter  equals  1  liter.  (The  liter  is  defined  as  the 
volume  occupied  by  the  mass  of  1  kilogram  of  water  under  a  pressure  of 
76  cm.  of  mercury  and  at  the  temperature  of  4  deg.  cent.  According  to  the 
best  information,  1  liter  =  1.000027  cubic  decimeters.) 

The  customary  weights  derived  from  the  international  kilogram  are  based 
on  the  value  1  avoirdupois  Ib.  =  453.59243  grams.  The  value  of  the  troy 
ib.  is  based  on  the  same  relation  and  also  the  equivalent  5760/7000 
avoirdupois  Ib.  equals  1  troy  Ib. 

Metric  Measures 


Length 

Area 

Unit 

Sym- 
bol 

Value  in  meters 

Unit 

Sym- 
bol 

Value  in  sq. 
meters 

Micron   

M 
mm. 
cm. 
dm. 
m. 
dkm. 
hm. 
km. 
Mm. 

0.000001 
0.001 
0.01 
0.1 
1.0 
10.0 
100.0 
1,000.0 
10,000.0 
1,000,000.0 

Millimeter.... 
Centimeter  .  . 
Decimeter.  .  . 
Meter  (unit). 
Dekameter.  . 
Hectometer.. 
Kilometer.  .  . 
Myriameter..    . 
Megameter..    . 

Sq.  millimeter  

mm.2 
cm.2 
dm.2 
m.» 
a. 
ha. 
km." 

0  .  000001 
0.0001 
0.01 
1.0 
100.0 
10,000.0 
1,000,000.0 

Sq.  centimeter  
Sq.  decimeter  

Sq.  meter  (centiare) 
Sq.  dekameter  (are) 
Hectare  

Sq.  kilometer.  . 

Volume 

Cubic  measure 

Unit 

Symbol 

Value  in 
liters 

Unit 

Symbol 

Value  in 
cubic 
meters 

Milliliter  
Liter  (unit) 

ml.  or  cm.3 
1.  or  dm.8 
kl.  or  m.s 

cl. 
dl. 
dkl. 
hi. 

0.001 
1.0 
1,000.0 

0.01 
0.1 
10.0 
100.0 

Cubic  kilometer  
Cubic  hectometer.  .  .  . 
Cubic  dekameter  

Cubic  meter  
Cubic  decimeter  
Cubic  centimeter  
Cubic  millimeter  
Cubic  micron  

km.3 
hm.» 
dkm.  s 

m.8 
dm.3 
cm.3 
mm.3 
M8 

10» 
10« 
103 

1 

10-  « 

10"  « 
ID'9 
10-18 

Kiloliter  
Also 
Centiliter  

Deciliter  

Dekaliter  

Hectoliter  

Weight 


Unit 

Symbol 

Value  in 
grams 

Unit 

Symbol 

Value  in 
grams 

Microgram  

0.000001 

Dekagram  

dkg. 

10.0 

Milligram 

mg. 

0.001 

Hectogram  

fag 

100  0 

Centigram  
Decigram           

eg- 
dg. 

0.01 
0.1 

Kilogram  
Myriagram  

kg. 
Mg 

1,000.0 
10,000  0 

Gram  (unit) 

1  0 

100  000  0 

Ton  

t. 

1,000,000.0 

SYSTEMS  OF  UNITS 

The  principal,  units  of  interest  to  mechanical  engineers  can  all  be  derived 
from  the  three  fundamental  units  of  force,  length,  and  time.  These  three 
fundamental  units  may  be  chosen  at  pleasure;  each  such  choice  gives  rise  to  a 
"system"  of  units.  The  following  table  gives  the  units  of  the  four  "systems" 
most  often  met  with  in  the  literature. 


UNITS 


73 


The  precise  definitions  of  the  fundamental  units  in  these  systems  are  as  follows.  (In 
these  definitions  the  "standard  pound  body  "  and  the  "standard  kilogram  body  "refer 
to  two  special  lumps  of  metal,  carefully  preserved  at  London  and  Paris,  respectively; 
the  "standard  locality"  means  sea  level,  45  deg.  latitude;  or,  more  strictly,  any  locality 
in  which  the  acceleration  due  to  gravity  has  the  value  980.665  cm.  per  sec.2  =  32.1740  ft. 
per  sec.2,  which  may  be  called  the  standard  acceleration. 

The  pound  (force)  is  the  force  required  to  support  the  standard  pound  body  against 
gravity,  in  vacua,  in  the  standard  locality;  or,  it  is  the  force  which,  if  applied  to  the  stand- 
ard pound  body,  supposed  free  to  move,  would  give  that  body  the  "standard  ac- 
celeration." The  word  "pound"  is  used  for  the  unit  of  both  force  and  mass,  and 
consequently  is  ambiguous.  To  avoid  uncertainty  it  is  desirable  to  call  the  units 
"pound  force"  and  "pound  mass,"  respectively. 

The  kilogram  (force)  is  the  force  required  to  support  the  standard  kilogram  against 
gravity,  in  vacua, in.  the  standard  locality;  or,  it  is  the  force  which,  if  applied  to  the  stand- 
ard kilogram  body,  supposed  free  to  move,  would  give  that  body  the  "standard  accelera- 
tion." The  word  "kilogram"  is  used  for  the  unit  of  both  force  and  mass  and  conse- 
quently is  ambiguous.  To  avoid  uncertainty  it  is  desirable  to  call  the  units  "kilogram 
force"  and  "kilogram  mass,"  respectively. 

The  poundal  is  the  force  which,  if  applied  to  the  standard  pound  body,  would  give 
that  body  an  acceleration  of  1  ft.  per  sec.2;  that  is,  1  poundal  =  1/32.1740  of  a  pound 
force. 

The  dyne  is  the  force  which,  if  applied  to  the  standard  gram  body,  would  give  that 
body  an  acceleration  of  1  cm.  per  sec.2;  that  is,  1  dyne  =  1/980.665  of  a  gram  force. 

Systems  of  Units 


British 

Metric 

Name  of 
unit 

Dimen- 
sions of 
units  in 
terms  of 
F,L,  T 

"gravita- 
tional "  sys- 
tem, or 
"foot-pound- 
second" 

"gravita- 
tional "  sys- 
tem, or 
"kilogram- 
meter-sec- 

Metric 
"absolute" 
system,  or 
"C.  G.  S." 
system 

British 
"absolute" 
system 
(little  used) 

system 

ond"  system 

Force  

F 

1  Ib. 

1  kg. 

1  dyne 

1  poundal 

Length  

L 

1  ft. 

1  m. 

1  cm. 

1ft. 

Time 

T 

1  sec. 

1  sec. 

1  sec. 

1  sec. 

Velocity  
Acceleration  .  . 

L/T 
L/T* 

1  ft.  per  sec. 
1  ft.  per  sec.2 

1  m.  per  sec. 
1  m.  per  sec.2 

1  cm.  per  sec. 
1  cm.  per  sec.2 

1  ft.  per  sec. 
1ft.  per  sec.2 

Pressure  

F/L* 

lib.  per  ft.2 

1  kg.  per  m.2 

1  dyne  per  cm.2 

1  pdl.  per  ft.2 

Impulse  or 

momentum.  . 

FT 

1  Ib.-sec. 

1  kg.-sec. 

1  dyne-sec. 

1  pdl.-sec. 

Work  or 

energy 

FL 

1  ft.-lb. 

1  kg.-m. 

1  dyne-cm.  = 

1  ft.  -pdl. 

1  "erg." 

Power  

FL/T 

1  ft.-lb.  per 

1  kg.-m.  per 

1  dyne-cm,  per 

1  ft.-pdl.  per 

sec. 

sec. 

sec. 

sec. 

Mass  

F/(L/T*) 

1  Ib.  per  (ft. 

1  kg.  per  (m. 

1  dyne  per  (cm. 

1  pdl.  per  (ft. 

per  sec.2)  = 

per  sec.2)  = 

per  sec.2)  =  1 

per    sec.2)  = 

1  "slug." 

1  "  metric 

gram  mass. 

1  pound 

slug." 

mass. 

NOTE.     The  "slug"  (also  called  the  "geepound,"  or  the  "engineer's  unit  of  mass"), 
the  "  metric  slug,"  and  the  "poundal"  are  never  used  in  practice. 

Other  common  units  are  as  follows: 
Work:       1  joule  =  10?  ergs  =  10,000,000  dyne-cm. 

1  kilowatt-hour  =  3,600,000  joules  =  3600  X  lO™  dyne-cm. 
Power:      1  horse  power  =  550  ft.-lb.  per  sec. 
1  poncelet  =  100  kg.-m.  per  sec. 
1  force  de  cheval  =  75  kg.-m.  per  sec. 
1  watt  =  1  joule  per  sec.  =  10,000,000  dyne-cm,  per  sec. 
1  kilowatt  =  1000  watts  =  1010  dyne-cm,  per  sec. 

A  new  horse  power  of  550.220  ft.-lb.  per  sec.,  or  746  watts,  has  been  proposed,  but  has 
not  been  accepted  by  mechanical  engineers. 

The  weight  of  a  body  (in  a  given  locality)  always  means  a  force,  namely,  the  force,  re- 


74 


WEIGHTS  AND  MEASURES 


quired  to  support  the  body  against  gravity  (in  that  locality).  'When  no  particular  local- 
ity is  specified,  the  standard  locality  may  be  assumed.  Thus,  the  "standard  weight"  of 
the  pound  body  is  1  lb.;  the  "standard  weight"  of  the  kilogram  body  is  1  kg 


Dynes  X  10«   Kilograms        Pounds        Poundala 


the  quantity  of  heat  required 
to  raise  the  temperature  of  1 
gram  of  water  1  deg.  cent,  at  a 
mean  temperature  of   15  deg. 
cent.,  or  (2)   the  heat  required 
to    raise    the    temperature    of 
1  lb.  of  water  1  deg.  fahr.     The 
former  quantity  is  called  the 
gram-calorie   (small  calorie), 
while  the  latter  is  known  as  the 
British  thermal  unitorB.t.u. 
The  kilogram-calorie  (large  calorie),  which  ia  equal  to  1000  g.-cal.,  is  largely 
used  in  engineering  work  in  metric  countries.  *    1  therm  =  1  g.-cal. 
CONVERSION  TABLES 
Length  Equivalents 


1 

1.020 

0.00848 

2.248  ' 
0.03518 

72.33 
1.85933 

0.9807 
1.99149 

1 

2.205 
0.34334 

70.93 

1.85C84 

0.4448 
1.64819 

0.4536 

1.65667 

1 

32.17 
1.50750 

0.01383 
2.14067 

0.01410 

2.14916 

0.03108 

2.49249 

1 

Centimeters       Inches 

Feet      |    Yards     |   Meters 

Chains 

Kilometers 

Miles 

1 

0.3937 
1.59517 

0.03281 

2.51598 

0.01094 
2.03886 

0.01 

2.00000 

0.03497I 
4.69644 

10-5 
B".  00000 

0.066214 
6.79335 

2.540 
0.40483 

1 

0.038333 

4.92082 

0.02778 
2.44370 

0.0254 

2.40483 

0.0*1263 

5.10127 

0.04254 

5.40483 

0.041578 

5.19818 

30.48 

1.48402 

12 

1.07918 

1 

0.3333 

T.52288 

0.3048 

1.48402 

0.01515 

2.18046 

0.033098 
4.48402 

0.0s  1645 

4.21608 

91.14 

1.96114 

36 

1.55630 

3 

0.47712 

1 

0.9144 
1.96114 

0.04545 

2.65758 

0.0s9144 
4.96114 

0.035682 
4.75449 

100 

2.00000 

39.37 

1.59517 

3.281 

0.51598 

1  .0936 

0.03886 

1 

0.04971 

2.69644 

0.001 
3.COOOO 

0.0362!4 

4.79335 

2012 
3.30356 

792 

2.89873 

66 

1.81954 

22 

1.34242 

20.12 
1.30356 

1 

0.02012 
2.30356 

0.0125 

2.09691 

100000 
5.00000 

39370 
4.59517 

3281 
3.51598 

1093.6 

3.C3886 

1000 

3.00000 

49.71 

1.69644 

1 

0.6214 
1.79335 

160925 
5.20665 

63360 

4.80182 

5280 
3.72263 

1760 
3.24551 

1609 
3.20665 

80 

1.90309 

1.609 

0.206G5 

1 

The  equivalents  are  given  in  the  heavier  type.     Logarithms  of  the  equivalents  are 
given  immediately  below. 

Subscripts  after  any  figure,  Os,  94,  etc.J  mean  that  that  figure  is  to  be  repeated  the 
indicated  number  of  times. 

Conversion  of  Lengths 


Inches 
to  milli- 
meters 

Milli- 
meters 
to  inches 

Feet 
to 
meters 

Meters 
to 
feet 

Yards 
to 

meters 

Meters 
to 
yards 

Miles 
to  kilo- 
meters 

Kilo- 
meters 
to  miles 

1 

3 
4 

6 
7 

[ 

25.40 
50.80 
76.20 
101  .60 

127.00 
152.40 
177.80 
203.20 
228.60 

0.03937 
0.07874 
0.1181 
0.1575 

0.1968 
0.2362 
0.2756 
0.3150 
0.3543 

0.3048 
0.6096 
0.9144 
1.219 

1.524 
1.829 
2.134 
2.438 
2.743 

3.281 
6.562 
9.842 
13.12 

16.40 
19.68 
22.97 
26.25 
29.53 

0.9144 
1.829 
2.743 
3.658 

4.572 
5.486 
6.401 
7.315 
8.230 

1.094 
2.187 
3.281 
4.374 

5.486 
6.562 
7.655 
8.749 
9.842 

1.609 
3.219 
4.828 
6.437 

8.047 
9.656    . 
11.27 
12.87 
14.48 

0.6214 
1.243 
1.864 
2.485 

3.107 
3.728 
4.350 
4.971 
5.592 

*See  Marks'  MECHANICAL  ENGINEERS'  HANDBOOK.. 


CONVERSION  TABLES 


75 


Mechanical  Equivalent  of  Heat.  See  p.  311.*  The  value  most  commonly 
accepted  among  American  engineers  as  the  work  equivalent  of  1  mean  B.t.u. 
is  777.5  ft.-lb.,  and  the  mean  gram-calorie  =  4.183  joules,  which  are  the 
values  used  throughout  this  book.  The  U.  S.  Bureau  of  Standards  does  not 
recommend  any  special  value;  for  its  own  purposes  it  takes  the  59  deg.  fahr. 
B.t.u.  as  778.2  ft.-lb.  and  the  68  deg.  B.t.u.  as  777.5  ft.-lb.  The  15  deg. 
calorie  =  4.187  joules;  20  deg.  calorie  =  4.183  joules.  There  is  an  uncer- 
tainty of  about  1  part  in  1000  in  these  values. 

Conversion  of  Lengths :  Inches  and  Millimeters 

Common  fractions  of  an  inch  to  millimeters 
(From  HU  to  1  in.) 


64ths 

Milli- 
meters 

64ths 

Milli- 
meters 

64ths 

Milli- 
meters 

64ths 

Milli- 
meters 

64ths 

Milli- 
meters 

64ths 

Milli- 
meters 

1 

0.397 

13 

5.159 

25 

9.922 

37 

14.684 

49 

19.447 

57 

22.622 

2 

0.794 

14 

5.556 

26 

10.319 

38 

15.081 

50 

19.844 

58 

23.019 

3 

1.191 

15 

5.953 

27 

10.716 

39 

15.478 

51 

20.241 

59 

23.416 

4 

1.588 

16 

6.350 

28 

11.113 

40 

15.875 

52 

20.638 

60 

23.813 

5 

1.984 

17 

6.747 

29 

11.509 

41 

16.272 

53 

21.034 

61 

24.209 

6 

2.381 

18 

7.144 

30 

11.906 

42 

16.669 

54 

21.431 

62 

24.606 

7 

2.778 

19 

7.541 

31 

12.303 

43 

17.066 

55 

21.828 

63 

25.003 

8 

3.175 

20 

7.938 

32 

12.700 

44 

17.463 

56 

22.225 

64 

25.400 

9 

3.572 

21 

8.334 

33 

13.097 

45 

17.859 

10 

3.969 

22 

8.731 

34 

13.494 

46 

18.256 

11 

4.366 

23 

9.128 

35 

13.891 

47 

18.653 

12 

4.763 

24 

9.525 

36 

14.288 

48 

19.050 

Decimals  of  an  inch  to  millimeters.     (From  0.01  in.  to  0.99  in.) 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

.0 

0.254 

0.508 

0.762 

1.016 

1.270 

1.524 

1.778 

2.032 

2.286 

.1 

2.540 

2.794 

3.048 

3.302 

3.556 

3.810 

4.064 

4.318 

4.572 

4.826 

.2 

5.080 

5.334 

5.588 

5.842 

6.096 

6.350 

6.604 

6.858 

7.112 

7.366 

.3 

7.620 

7.874 

8.128 

8.382 

8.636 

8.890 

9.144 

9.398 

9.652 

9.906 

.4 

10.160 

10.414 

10.668 

10.922 

11.176 

11.430 

11.684 

11.938 

12.192 

12.446 

.5 

12.700 

12.954 

13.208 

13.462 

13.716 

13.970 

14.224 

14.478 

14.732 

14  986 

.6 

15.240 

15.494 

15.748 

16.002 

16.256 

16.510 

16.764 

17.018 

17.272 

17.526 

.7 

17.780 

18.034 

18.288 

18.542 

18.796 

19.050 

19.304 

.  19.558 

19.812 

20.066 

.8 

20.320 

20.574 

20.828 

21  .082 

21.336 

21.590 

21  .844 

22.098 

22.352 

22.606 

.9 

22.860 

23.114 

23.368 

23.622 

23.876 

24.130 

24.384 

24.638 

24.892 

25.146 

Millimeters  to  decimals  of  an  inch.      (From  1  to  99  mm.) 


0. 

1. 

2. 

3. 

4. 

5. 

6. 

7. 

8. 

9. 

0 

0.0394 

0.0787 

0.1181 

0.1575 

0.1969 

0.2362 

0.2756 

0.3150 

0.3543 

1 

0.3937 

0.4331 

0.4724 

0.5118 

0.5512 

0.5906 

0.6299 

0.6693 

0.7087 

0.7480 

2 

0.7874 

0.8268 

0.8661 

0.9055 

0.9449 

0.9843 

1  .0236 

1  .0630 

1.1024 

1.1417 

3 

1.1811 

1.2205 

1  .2598 

1.2992 

1.3386 

1.3780 

1.4173 

1.4567 

1  .4961 

1.5354 

4 

1.5748 

1.6142 

1  .6535 

1  .6929 

1.7323 

1.7717 

1.8110 

1.8504 

1.8898 

1.9291 

5 

1.9685 

2.0079 

2.0472 

2.0866 

2.1260 

2.1654 

2.2047 

2.2441 

2.2835 

2.3228 

6 

2.3622 

2.4016 

2.4409 

2.4803 

2.5197 

2.5591 

2.5984 

2.6378 

2.6772 

2.7165 

7 

2.7559 

2.7953 

2.8346 

2.8740 

2.9134 

2.9528 

2.9921 

3.0315 

3.0709 

3.1102 

8 

3.1496 

3.1890 

3.2283 

3.2677 

3.3071 

3.3465 

3.3858 

3.4252 

3.4646 

3.5039 

9 

3.5433 

3.5827 

3.6220 

3.6614 

3.7008 

3.7402 

3.7795 

3.8189 

3.8583 

3.8976 

'See  Marks'  MECHANICAL  ENGINEERS'  HANDBOOK. 


76 


WEIGHTS  AND  MEASURES 


Area  Equivalents 

(For  conversion  table  see  p.  77) 


Square 
meters 

Square 
inches 

Square 
feet 

Square 
yards 

Square 
rods 

Square 
chains 

Roods 

Acres 

Square 
miles  or 
sections 

1 

1550 

3.19033 

10.76 
1.03197 

1.196 
0.07773 

0.0395 

2.59699 

0.002471 

3.39288 

0.039884 
3.99494 

0.0>2471 

1.39288 

0.0*3861 

7.58670 

0.036452 
4  80967 

1 

0.006944 
3.84164 

0.0011 

3.88740 

0.042551 

5.40667 

0.0*1594 
6.20255 

0  066377 

7.80461 

0.0e1594 

7.20255 

0.084910 

10.39637 

0.09290 
1.96803 

144 
2.15836 

1 

0.1111 

1.04576 

0.003673 
3.56503 

0.032296 
4.36091 

0.049184 

5.96297 

0.042296 
4.36091 

0.0:3587 

"S.  554  73 

0.8361 

1.92227 

1296 
3.11260 

9 

0.95424 

1 

0.03306 

2.51927 

0.002066 
3.31515 

0.038264 
4.91721 

0.0002066 
4.31515 

0.063228 

7.50898 

25.29 
1.40300 

39204 
4.59333 

272.25 
2.43497 

30.25 

1.48072 

1 

0.0625 

2.79588 

0.02500 

2.39794 

0.00625 

3.79588 

0.  0*9766 

6.98970 

404.7 
2.60712 

627264 

5.79745 

4356 
3.63909 

484 

2.68484 

16 

1.20412 

1 

0.4 

1.60206 

0.1 

1.00000 

0.0001562 

4.19382 

1012 

3.00506 

1568160 
6.19539 

10890 
4.03703 

1210 

3.08278 

40 

1.60206 

2.5 

0.39794 

1 

0.25 

1.39794 

0.033906 
"4.59176 

4047 
3.60712 

6272640 
6.79745 

43560 

4.63909 

4840 

3.68484 

160 

2.20412 

10 

1.00000 

4 

0.60206 

1 

0.001562 
3.19382 

2589a8 
6.41330 

27878400 

7.44527 

3097600 
6.49102 

102400 
5.01030 

6400 

3.80618 

2560 

3.40824 

640 

2.80618 

1 

(1  hectare  =100  ares  =  10,000  centiares  or  square  meters) 

Volume  and  Capacity 'Equivalents 

(For  conversion  table  see  p.  77) 


Cubic 
inches 

Cubic 
feet 

Cubic 
yards 

U.  S. 
Apothe- 
cary 
liquid 
ounces 

U.   S.   quarts 

U.  S.  gallons 

Bushels 
U.  S. 

Liters 
(1) 

Liquid 

«ry 

Liquid 

Dry 

1 

0.035787 

4.76246 

0.042143 
F.  33109 

0.5541 

1.  74360 

0.01732 
2.23845 

0.01488 

2.17263 

0.024329 
3.63639 

0.023720 

3.57057 

0.034650 

4.66748 

0.01639 
2.21450 

1728 
3.23754 

1 

0.03704 

2".  56864 

957.5 
2.98114 

29.92 

1.47599 

25.71 

1.41017 

7.481 
0.87393 

6.429 

0.80811 

0.8036 

1.90502 

28.32 
1.45205 

46656 
4.66891 
1.805 
0.25640 

27      . 

1.43136 
0  001044 

3.01886 

1 

0.043868 

5.58749 

25853 
4.41251 

1 

807.9 
2.90736 
0.03125 

2.49485 

694.3 
2.84153 
0.02686 

2.42903 

202.0 
2.30530 
0.007813 
3.89279 

173.6 

2.23948 
0.006714 

3.82697 

21.70 
1.33638 
0.038392 

4.92388 

764.6 
2.88341 
0.02957 

2.47091 

57.75 

1.76155 

0.03342 

2.52401 

0.001238 
3.09264 

32 
1.50515 

1 

0.8594 
1.93418 

0.25 

1.39794 

0.2148 
1.33212 

0.02686 
2.42903 

0.9464 
1.97606 

67.20 
1.82737 

0.03889 
2.58983 

0.001440 
3.15847 

37.24 

1.57097 

1.164 

0.06582 

1 

0  2909 
1.46376 

0.25 

1.39794 

0.03125 

2.49485 

1.101 

0.04188 

231 
2.36361 

0.1337 

1.12607 

0.004951 
3.69470 

128 

2.10721 

4 
0.60206 

3.437 

0.53624 

1 

0.8594 
1.93418 

0.1074 
1.03109 

3.785 

0.57812 

268.8 
2.42943 

0.1556 
1.19189 

0.005761 

3.76053 

148.9 
2.17303 

4.655 

0.66788 

4 

0.60206! 

1.164 

0.06582 

1 

0.125 

1.09691 

4.405 
0.64394 

2150 
3.33252 

1.244 
0.09498 

0.04609 
2.66362 

1192 
3.07612 

37.24 

1.57097 

32 
1.50515 

9.309 

0.96891 

8 

0.90309 

1 

35.24 
1.54703 

61.02 
1.78550 

0.03531 
2.54795 

0.001308 
3.11659 

33.81 

1.52909 

1.057 

0.02394 

0.9081 
1.95812 

0.2642 

L  42188 

0.2270 
1.35606 

0  02838 
2.45297 

1 

The  equivalents  are  given  in  the  heavier  type.  Logarithms  of  the  equivalents  are 
given  immediately  below. 

Subscripts  after  any  figure,  Oa,  94,  etc.,  mean  that  that  figure  ia  to  be  repeated  the 
indicated  number  of  timeB. 


CONVERSION  TABLES 


77 


Conversion  of  Areas 


Sq.  in. 
to 
sq.  cm. 

Sq.  cm. 
to 
sq.  in. 

Sq.  ft. 
to 
sq.  m. 

Sq.  m. 
to 
sq.  ft. 

Sq.  yd. 
to 
sq.  m. 

Sq.  m. 
to 
sq.  yd. 

Acres 
to 
hec- 
tares 

Hec- 
tares to 
acres 

Sq.  mi. 
to 
sq.  km. 

Sq.  km. 
to 
sq.  mi. 

1 

2 
3 

4 

5 
6 
7 
8 
9 

6.452 
12.90 
19.35 
25.81 

32.26 
38.71 
45.16 
51.61 
58.06 

0.1550 
0.3100 
0.4650 
0.6200 

0.7750 
0.9300 
1.085 
1.240 
1.395 

0.0929 
0.1858 
0.2787 
0.3716 

0.4645 
0.5574 
0.6503 
0.7432 
0.8361 

10.76 
21.53 
32.29 
43.06 

53.82 
64.58 
75.35 
86.11 
96.87 

0.8361 
1.672 
2.508 
3.345 

4.181 
5.017 
5.853 
6.689 
7.525 

1.196 
2.392 
3.588 
4.784 

5.980 
7.176 
8.372 
9.568 
10.764 

0.4047 
0.8094 
1.214 
1.619 

2.023 
2.428 
2.833 
3.237 
3.642 

2.471 
4.942 
7.413 
9.884 

12.355 
14.826 
17.297 
19.768 
22.239 

2.590 
5.180 
7.770 
10.360 

12.950 
15.540 
18.130 
20.720 
23.310 

0.3861 
0.7722 
1.158 
1.544 

1.931 
2.317 
2,703 
3.089 
3.475 

Conversion  of  Volumes  or  Cubic  Measure 


Cu.  in. 
to 
cu.  cm. 

Cu.  cm. 
to 
cu.  in. 

Cu.  ft. 
to 
cu.  m. 

Cu.  m. 
to 
cu.  ft. 

Cu.  yd. 
to 
Cu.  m. 

Cu.  m. 
to 
cu.  yd. 

Gallons 
to 
cu.  ft. 

Cu.  ft. 
to 
gallons 

2 
3 
4 

5 
6 
7 

8 
9 

16.39 
32.77 
49.16 
65.55 

81.94 
98.32 
114.7 
131.1 
147.5 

0.06102 
0.1220 
0  1831 
0.2441 

0.3051 
0.3661 
0.4272 
0.4882 
0.5492 

0.02832 
0.05663 
0.08495 
0.1133 

0.1416 
0.1699 
0.1982 
0.2265 
0.2549 

35.31 
70.63 
105.9 
141.3 

176.6 
211.9 
247.2 
282.5 
317.8 

0.7646 
1.529 
2.294 
3.058 

3.823 
4.587 
5.352 
6.116 
6.881 

1.308 
2.616 
3.924 
5.232 

6.540 
7.848 
9.156 
10.46 
11.77 

0.1337 
0.2674 
0.4011 
0.5348 

0.6685 
0.8022 
0.9359 
1.070 
1.203 

7.481 
14.96 
22.44 
29.92 

37.41 
44.89 
52.36 
59.85 
67.33 

Conversion  of  Volumes  or  Capacities 


Liquid 
ounces 
to 
cu.  cm. 

Cu.  cm. 
to 
liquid 
ounces 

Pints 
to 
liters 

Liters 
to 
pints 

Quarts 
to 
liters 

Liters 
to 
quarts 

Gallons 
to 
liters 

Liters 
to 
gallons 

Bushels 
to 
hecto- 
liters 

Hecto- 
liters 
to 
bushels 

1 

29.57 

0.03381 

0.4732 

2.113 

0.9464 

1.057 

3.785 

0.2642 

"0.3524 

2.838 

2 

59.15 

0.06763 

0.9464 

4.227 

1.893 

2.113 

7.571 

0.5283 

0.7048 

5.676 

3 

88.72 

0.1014 

1.420 

6.340 

2.839 

3.170 

11.36 

0.7925 

1.057 

8.513 

4 

118.3 

0.1353 

1.893 

8.453 

3.785 

4.227 

15.14 

1  .057 

1.410 

11.35 

5 

147.9 

0.1691 

2.366 

10.57 

4.732 

5.283 

18.93 

1.321 

1.762 

14.19 

6 

177.4 

0.2029 

2.839 

12.68 

5.678 

6.340 

22.71 

1.585 

2.114 

17.03 

7 

207.0 

0.2367 

3.312 

14.79 

6.625 

7.397 

26.50 

1.849 

2.467 

19.86 

8 

236.6 

0.2705 

3.785 

16.91 

7.571 

8.453 

30.28 

2.113 

2.819 

22.70 

9 

266.2 

0.3043 

4.259 

19.02 

8.517 

9.510 

34.07 

2.378 

3.172 

25.54 

Conversion  of  Masses 


Grains 
to 

grams 

Grams 
to 
grains 

Ounces 
(avoir.) 
to 
grams 

Grams 
to 
ounces 
(avoir.) 

Pounds 
(avoir.) 
to 
kilo- 
grams 

Kilo- 
grams 
to 
pounds 
(avoir.) 

Short 
tons 
(2000 
Ib.)  to 
metric 
tons 

Metric 
tons 
(1000 
kg.)  to 
short 
tons 

Long 
tons 
(2240 
Ib.)  to 
metric 
tons 

Metric 
tons 
to 
long 
tons 

1 

0.06480 

15.43 

28.35 

0.03527 

0.4536 

2.205 

0.907 

1.102 

1.016 

0.984 

2 

0.1296 

30.86 

56.70 

0.07055 

0.9072 

4.409 

1.814 

2.205 

2.032 

1.968 

3 

0.1944 

46.30 

85.05 

0.1058 

1.361 

6.614 

2.722 

3.307 

3.048 

2.953 

4 

0.2592 

61.73 

113.40 

0.1411 

1.814 

8.818 

3.629 

4.409 

4.064 

3.937 

5 

0.3240 

77.16 

141.75 

0.1764 

2.268 

11.02 

4.536 

5.512 

5.080 

4.921 

6 

0.3888 

92.59 

170.10 

0.2116 

2.722 

13.23 

5.443 

6.614 

6.096 

5.905 

7 

0.4536 

108,03 

198.45 

0.2469 

3.175 

15.43 

6.350 

7.716 

7.112 

6.889 

8 

0.5184 

123.46 

226.80 

0.2822 

3.629 

17.64 

7.257 

8.818 

8.128 

7.874 

9 

0.5832 

138.89 

255.15 

0.3175 

4.082 

19.84 

8.165 

9.921 

9.144 

8.857 

78 


WEIGHTS  AND  MEASURES 


Velocity  Equivalents 

(For  conversion  table  see  p.  80) 


Centimeters 
per  sec. 

Meters 
per  sec. 

Meters 
per  min. 

Kilo- 
meters 
per  hour 

Feet 
per  sec. 

Feet 
per  min. 

Miles 
per  hour 

Knots 

1 

0.01 

0.6 

1.77815 

0.036 

2.55630 

0.03281 
2.51598 

1.9685 
0.29414 

0.02237 

2.34965 

0.01942 

2.28825 

100 

2.00000 

1 

60 
1.77815 

3.6 

0.55630 

3.281 

0.51598 

196.85 

2.29414 

2.237 

0.34965 

1.942 

0.28825 

1.667 
0.22184 

0.01667 
2.22184 

1 

0.06 

2.77815 

0.05468 

2.73783 

3.281 
0.51598 

0.03728 
2.57150 

0.03237 

2.51018 

27.78 
1.44370 

0.2778 
1.  44370 

16.67 

1.22184 

1 

0.9113 

T.  95968 

54.68 
1.73783 

0.6214 

1.79335 

0.53960 

1.73207 

30.48 
1.48402 

0.3048 

F.  48402 

18.29 

1.26217 

1.097 

0.04032 

1 

60 

1.77815 

0.6818 

1.83367 

0.59209 

1.77238 

0.5080 
1.70586 

0.005080 
3".  70586 

0.3048 

1.48402 

0.01829 
2.26217 

0.01667 

2.22185 

1 

0.01136 

2.05553 

0.00987 
3.99423 

44.70 
1.65035 

0.4470 

r.  65035 

26.82 
1.42850 

1.609 

0.20670 

1.467 
0.16633 

88 
1.94448 

1 

0.86839 
1.93871 

51.497 

1.71178 

0  51497 
F.  71178 

30.898 
1.48993 

1.8532 
0.26793 

1.68894 
0.22761 

101.337 

2.00577 

1.15155 

0.06128 

1 

Mass  Equivalents 

(For  conversion  table  see  p.  77) 


Kilograms 

Grains 

Ounces 

Pounds 

Tons 

Troy  and 
apoth. 

Avoir- 
dupois 

Troy  and 
apoth. 

Avoir- 
dupois 

Short 

Long 

Metric 

1 

15432 
4.18843 

32.15 
1.50719 

35.27 
1.54745 

2.6792 
0.42801 

2.205 
0.34333 

0.021102 
3.04230 

0'039842 
4.99309 

0.001 
3.  00000 

0.046480 
5.81157 

1 

0.022083 
3.31876 

0.022286 
3.35902 

0.031736 

4.23958 

0.031429 
4.15490 

0.077143 

"8.  85387 

0.076378 

8.80465 

0.0:6480 

8.81157 

0.03110 

2.49281 

480 
2.68124 

1 

1.09714 
0.04026 

0.08333 

2.92082 

0.06857 
2.83614 

0.043429 
5.53511 

0.043061 

5.48590 

0.043110 

"5.49281 

0.02835 
2.45255 

437.5 

2.64098 

0.9115 

1.95974 

1 

0.07595 

2.88056 

0.0625 

2.79588 

0.043125 

5.49485 

0.042790 
5.44563 

0.042835 
T.  45255 

0.3732 
T.57199 

5760 

3.76042 

12 

1.07918 

13.17 

1.11944 

1 

0.8229 
1.91532 

8.0*41  H 

4.61429 

0.0s3673 

4.56508 

0.033732 
4.57199 

0.4536 
T.  65667 

7000 
3.84510 

14.58 
1.16386 

16 

1.20412 

1.215 

0.08468 

1 

0.0005 

T.  69897 

0.034464 
4.64975 

0.034536 

4.65667 

907.2 
2.95770 

140e 

7.14613 

29167 
4.46489 

3203 
4.50515 

2431 
3.38571 

2000 
3.30103 

1 

0.8929 
T.  95078 

0.9072 
1.95770 

1016 
3.00691 

156804 
7.19535 

326s 
4.51411 

35840 
4.55437 

2722 
3.43492 

2240 
3.35025 

1.12 

0.04922 

1 

1.016 

0.00691 

1000 
3.00000 

15432356 

7.18843 

32151 
4.50719 

35274 
4.54745 

2679 
3.42801 

2205 
3.34333 

1.102 

0.04230 

0.9842 
1.99309 

1 

The  equivalents  are  given  in  the  heavier  type.  Logarithms  of  the  equivalents  are 
given  immediately  below. 

Subscripts  after  any  figure,  Os,  §4, -etc.,  mean  that  that  figure  is  to  be  repeated  the 
indicated  number  of  times. 


CONVERSION  TABLES 


79 


Pressure  Equivalents 

(For  conversion  table  see  p.  80) 


Megabars 
or 
megadynes 
per 
sq.  cm. 

Kilo- 
grams 
per 
sq.  cm. 
(Metric 
atmos- 
pheres) 

Pounds 
per 
sq.  in." 

Short 
tons 
per 

sq.  ft. 

Atmos- 
pheres 

Columns  of 
mercury  at 
temperature 
0°C. 

Columns  of  water  at 
temperature    15°    C. 

Meters 

Inches 

Meters 

Inches 

Feet 

1 

1.0197 

14.50 

1.044 

0.9S69 

0.7500 

29.53 

10.21 

401.8 

33.48 

0.00848 

1.16148 

0.01882 

1.99427 

1.87508 

1.47025 

1.00886 

2.60402 

1.52484 

0.9807 

1 

14.22 

1.024 

0.9678 

0.7355 

28.96 

10.01 

394.0 

32.84 

1.99152 

1.15300 

0.01034 

1.98579 

1.86660 

1.46177 

1.00038 

2.59555 

1.51636 

0.06895  - 

0.07031 

1 

0.072 

0.06804 

0.05171 

2.036 

0.7037 

27.70 

2.309 

2.83852 

2.84700 

2".  85733 

2.83279 

2.71360 

0.30876 

f.  84738 

1.44254 

0.36336 

0.9576 

0.9765 

13.89 

1 

0.9450 

0.7182 

28.28 

9.773 

384.8 

32.06 

,1.98119 

1.  98966 

1.14267 

1.97545 

1.85627 

1.45143 

0.99004 

2.58521 

1.50603 

1.0133 

1.0333 

14.70 

1.058 

1 

0.76 

29.92 

10.34 

407.2 

33.93 

0.00573 

0.01421 

1.16722 

0.02955 

1.88081 

1.47598 

1.01459 

2.60976 

1.53058 

1.3333 

1.3596 

19.34 

1.392 

1.316 

1 

39.37 

13.61 

535.7 

44.64 

0.12492 

0.13340 

1.28640 

0.14373 

0.11919 

1.59517 

1.13378 

2.72894 

1.64976 

0.03386 

0.03453 

0.4912 

0.03536 

0.03342 

0.02540 

1 

0.3456 

13.61 

1.134 

2".  52975 

2.53823 

T.  69124 

2.54857 

2.52402 

2.40484 

1.53861 

1.13378 

0.05460 

0.09798 

0.09991 

1.421 

0.1023 

0.09670 

0.07349 

2.893 

| 

39.37 

3.281 

2~.99114 

2".  99962 

0.15262 

F.  00996 

2.98541 

2.86622 

0.46139 

1.59517 

0.55198 

0.002489 

0  002538 

0.03610 

0.002599 

0.002456 

0.001867 

0.07349 

0.02540 

1 

0.08333 

3.39598 

3.40446 

2.55746 

3.41479 

3.39024 

3.27106 

2.86622 

2~.40484 

2.92082 

0.02986 

0.03045 

0.4332 

0.03119 

0.02947 

0.02240 

0.8819 

0.3048 

12 

1 

2.47516 

2.48364 

1.63664 

2.49397 

2.46942 

2.35024 

1.94540 

1.48402 

1.07918 

Energy  or  Work  Equivalents 

(For  conversion  table  see  p.  80) 


Joules  = 
10'  ergs 

Kilogram- 
meters 

Foot- 
pounds 

Kilo- 
watt- 
hours 

Cheval- 
vapeur- 
hours 

Horse- 
power- 
hours 

Liter- 
atmos- 
pheres 

Kilo- 
gram- 
calories 

British 
thermal 
units 

1 

0.10197 

1.00848 

0.7376 

1.86780 

0.0e2778 
7.44370 

0.0o3777 
7.57711 

0.063725 
7.57113 

0.009869 
3.99427 

0.0»2390 

T.  37848 

0.039486 
4.97709 

9.80665 
0.9915207 

1 

7.233 
0.85932 

0.052724 
6.43522 

0.0837037 

6.56863 

0.053653 

6.56265 

0.09678 

2.98579 

0.002344 
3.37000 

0  009302 
3.96861 

1.356 
0.13220 

0.1383 
1.14068 

1 

0.063766 
7.57590 

0.0651206 

7.70932 

0.0650505 
7.70333 

0.01338 

2.12647 

0.033241 

4.51068 

0.001286 

3.10929 

3.6X10« 

6.55630 

3.671X105 

5.56478 

2.655X10" 
6.42410 

'" 

1.3596 
0.13342 

1.341 
0.12743 

35528 
4.55057 

860.5 

2.93478 

3415 
3.53339 

2.648X10« 
6.42288 

270000. 
5.43136 

1.9529X10* 
6.29068 

0.7355 

1.86658 

1 

0.9863 
1.99401 

26131. 

4.41715 

632.9 

2.80135 

2512 

3.39996 

2.6845X108 

6.42887 

2.  7375X105 
5.43735 

1.98X10« 
6.29667 

0.7457 

1.87257 

1.0139 
0.00598 

1 

26494 
4.42314 

641.7 

2.80735 

2547 
3.40595 

101.33 

2.00573 

10.333 
1.01421    ' 

74.73 

1.87353 

0.042815 
5.44943 

0.043827 

5.58284 

0.043774 

5.57686 

1 

0.02422 
2.38425 

0.09612 

2.98281 

4183 
3.62153 

426.6 
2.63000 

3086 
3.48932 

0.001162 

3.06522 

0.001580 

3.19864 

0.001558 
3.19265 

41.29 
1.61579 

1 

3.968 
0.59861 

1054 

3.02291 

107.5 
2.03139 

777.52 
2.89071 

0.032928 

4.46661 

0.033981 

4.60003 

0.033927 

4.59405 

10.40 
1.01719 

0.25200 
1.40139 

1 

The  equivalents  are  given  in  the  heavier  type.  Logarithms  of  the  equivalents  are 
given  immediately  below. 

Subscripts  after  any  figure,  Oi,  94,  etc.,  mean  that  that  figure  is  to  be  repeated  the 
indicated  number  of  times. 

V 


80 


WEIGHTS  AND   MEASURES 


Linear   and   Angular  Velocity    Conversion   Factors 


Cm.  per 
sec.  to 
feet  per 
min. 

Feet  per 
min.  to 
cm.  per 
sec. 

Cm.  per 
sec.  to 
miles 
per  hour 

Miles 
per  hour 
to  cm. 
per  sec. 

Feet  per 
sec.   to 
miles 
per  hour 

Miles 
per  hour 
to  feet 
per  sec. 

Radians 
per  sec. 
to  rev. 
per  min. 

Rev.  per 

min.  to 
radians 
per  sec. 

1 

2 
3 

4 

5 
6 
7 

8 
9 

1.97 
3.94 
5.91 
7.87 

9.84 
11.81 
13.78 
15.75 
17.72 

0.508 
1.016 
1.524 
2.032 

2.540 
3.048 
3.556 
4.064 
4.572 

0.0224 
0.0447 
0.0671 
0.0895 

0.1118 
0.1342 
0.1566 
0.1789 
0.2013 

44.7 
89.4 
134.1 
178.8 

223.5 
268.2 
312.9 
357.6 
402.3 

0.682 
1.364 
2.046 
2.727 

3.409 
4.091 
4.773 
5.455 
6.136 

1.47 
2.93 
4.40 
5.87 

7.33 
8.80 
10.27 
11.73 
13.20 

9.55 
19.10 
28.65 
38.20 

47.75 
57.30 
66.85 
76.39 
85.94 

0.1047 
0.2094 
0.3142 
0.4189 

0.5236 
0.6283 
0.7330 
0.8378 
0.9425 

Conversion  of  Pressures 


Pounds  per 
sq.  in.  to 
kilograms 
per  sq.  cm. 

Kilograms 
per  sq.  cm. 
to  pounds 
per  sq.  in. 

Atmospheres 
to  pounds 
per  sq.  in. 

Pounds  per 
sq.  in.  to 
atmospheres 

Atmospheres 
to  kilograms 
per  sq.  cm. 

Kilograms 
per  sq.  cm. 
to  atmos- 
pheres 

1 

0.0703 

14.22 

14.70 

0.0680 

1.033 

0.9678 

2 

0.1406 

28.45 

29.39 

0.1361 

2.067 

1.936 

3 

0.2109 

42.67 

44.09 

0.2041 

3.100 

2.903 

4 

0.2812 

56.89 

58.79 

0.2722 

4.133 

3.871 

5 

0.3515 

71.12 

73.48 

0.3402 

5.166 

4.839 

6 

0  4218 

85.34 

88.18 

0.4082 

6.200 

5.807 

7 

0.4922 

99.56 

102.9 

0.4763 

7.233 

6.774 

8 

0.5624 

113.8 

117.6 

0.5443 

8.266 

•7.742 

9 

0.6328 

128.0 

132.3 

0.6124 

9.300 

8.710 

Conversion  of  Energy,  Work,  Heat 


Ft.-lb. 
to 
kilo- 
gram- 
meters 

Kilo- 
gram- 
meters 
to 
ft.-lb. 

Ft.-lb. 
to 
B.t.u. 

B.t.u. 
to 
ft.-lb. 

Kilo- 
gram- 
meters 
to 
large 
calories 

Large 
calories 
to 
kilo- 
gram- 
meters 

Joules 
to 
small 
calories 

Small 
calories 
to 
j  oules 

1 

0.1383 

7.233 

0.001286 

777.5 

0.002344 

426.6 

0.2390 

4.183 

2 

0.2765 

14.47 

0.002572 

1555.0 

0.004688 

853.2 

0.4780 

8.367 

3 

0.4148 

21.70 

0  003858 

2333.0 

0007033 

1280.0 

0'.7170 

12.55 

4 

0.5530 

28.93 

0.005144 

3110.0 

0.009377 

1706.0 

0.9560 

16.73 

5 

0.6913 

36.16 

0.006431 

3888.0 

0.01172 

2133.0 

1.195 

20.92 

6 

0.8295 

43.40 

0007717 

4665.0 

0.01407 

2560.0 

1.434 

25.10 

7 

0.9678 

50.63 

0009003 

5443.0 

0.01641 

2986.0 

1.673 

29.28 

8 

1.106 

57.86 

0  01029 

6220.0 

0.01875 

3413.0 

1.912 

33.47 

9 

1.244 

65.10 

0.01157 

6998.0 

0.02110 

3839.0 

2.151 

37.65 

Conversion  of  Power 


Horse  powers 
to  kilowatts 

Kilowatts  to 
horse  powers 

Metric 
horse  powers 
to  kilowatts 

Kilowatts 
to    metric 
horse  powers 

Horse  powers 
to    metric 
horse  powers 

Metric 
horse  powers 
to 
horse  powers 

1 

0.7457 

1.341 

0.7354 

1.360 

1.014 

0.9863 

2 

1.491 

2.682 

1.471 

2.719 

2.028 

1.973 

3 

2.237 

4.023 

2.206 

4.079 

3.042 

2.959 

4 

2.983 

5.364 

2.942 

5.439 

4.056 

3.945 

5 

3.728 

6.705 

3.677 

6.799 

5.069 

4.932 

6 

4.474 

8.046 

4.413 

8.158 

6.083 

5.918 

7 

5.220 

9.387 

5.148 

9.518 

7.097 

6.904 

8 

5.965 

10.73 

5.884 

10.88 

8.111 

7.890 

9 

6.710 

12.07 

6.619 

12.24 

9.125 

8.877 

CONVERSION  TABLES 


81 


Power  Equivalents 

(For  conversion  table  see  p.  80) 


Horse  power 

Kilo- 
watts 
(1000 
joules 
per  sec.) 

Cheval- 
vapeur 
(metric 
h.p.) 

Ponce- 
lets 

M.-kg. 
per  sec. 

Ft.-lb. 
per  sec. 

Kg- 
cal. 
per  sec. 

B.t.u 
per  sec. 

550  stand- 
ard ft.-lb. 
per  sec. 

1 

0.7457 

1.014 

0.7604 

76.04 

550 

0.1783 

0.7074 

1.87256 

0.00599 

1.88105 

1.88105 

2.74036 

1.25104 

1.84965 

1.341 

1 

1.360 

1.020 

102.0 

737.6 

0.2390 

0.9486 

0.12743 

0.13343 

0.00848 

2.00848 

2.86780 

1.37848 

1.97709 

0.9863 

0.7355 

1 

0.75 

75 

542.3 

0.1758 

0.6977 

T.  99402 

T.  86659 

1.87506 

1.87506 

2.73438 

1.24506 

1.84367 

1.315 

0.9807 

1.333 

1 

100 

723.3 

0.2344 

0.9303 

0.11896 

1.99152 

0.12493 

2.00000 

2.85932 

1.37000 

1.96861 

0.01315 

0.009807 

0.01333 

0.01 

1 

7.233 

0.002344 

0.009303 

2.11896 

3.99152 

2.12493 

T.  00000 

0.85932 

3.37000 

2.96861 

0.00182 

0.001356 

0.00184 

0.00138 

0.1383 

1 

0.033241 

0.001286 

3.25946 

3.13219 

3.26562 

3.14067 

T.  14067 

T.  51068 

T.  10929 

5.610 

4.183 

5.688 

4.266 

426.6 

3086 

1 

3.968 

0.74896 

0.62153 

0.75494 

0.63000 

2.63000 

3.48932 

0.59861 

1.414 

1.054 

1.433 

1.075 

107.5 

777.5 

0.2520 

1 

0.15035 

0.02291 

0.15632 

0.03139 

2.03139 

2.89071 

1.40138 

The  equivalents  are  given  in  the  heavier  type.  Logarithms  of  the  equivalents  are 
given  immediately  below. 

Subscripts  after  any  figure,  Os,  94,  etc.,  mean  that  that  figure  is  to  be  repeated  the 
indicated  number  of  times. 

Density  Equivalents  and  Conversion  Factors 


Equivalents 


Conversion  factors 


Grams 
per  cu. 
cm. 

Lb.  per 

cu.  in. 

Lb.  per 
cu.  ft. 

Short 
tons 
(2000 
lb.)  per 
cu.  yd. 

Lb.  per 
U.  S. 
gal. 

Grams 
per  cu. 
cm.  to 
lb.  per 
cu.   ft. 

Lb.  per 

cu.  ft. 
to   grams 
per  cu. 
cm. 

Grams 
per 
cu.  cm. 
to  short 
tons  per 
cu.  yd. 

Short 
tons  per 
cu.  yd. 
to  grams 
per  cu. 
cm. 

1 

0.03613 

62.43 

0.8428 

8.345 

1 

62.43 

0.01602 

0.8428 

1.186 

2.55787 

1.79539 

1.92572 

0.92143 

2 

124.90 

0.03204 

1.6860 

2.373 

27.68 

1 

1728 

23.33 

231 

3 

187.30 

0.04806 

2.5280 

3.600 

1.44217 

3.23754 

1.36792 

2.36361 

4 

249.70 

0.06407 

3.3710 

4.746 

0.01602 

0.035787 

1 

0.0135 

0.1337 

5 

312.40 

0.08009 

4.2140 

5.933 

2.20466 

4.76245 

2.13033 

1.12613 

6 

374.60 

0.09611 

5.0570 

7.119 

1.186 

0.04286 

74.07 

1 

9.902 

7 

437.00 

0.11210 

5.9000 

8.306 

0.07428 

2.63205 

1.86964 

0.99572 

8 

499.40 

0.12820 

6.7420 

9.492 

0.1198 

0.004329 

7.481 

0.1010 

1 

9 

561.90 

0.14420 

7.5850 

10.680 

1.07855    3.63639 

0.87396 

1.00432 

10 

624.30 

0.16020 

8.4280 

11.870 

82 


WEIGHTS  AND  MEASURES 


Conversion  of  Heat  Transmission  and  Conduction 


Small 

B.t.u. 

Small 

B.t.u. 

Small  calories  per 

B.t.u.  per  hr.  per 

calories 

per  sq. 

calories 

per  sq.  ft. 

sec.  per  sq.  cm. 

sq.  ft.  per  1  deg. 

per  sq. 
cm.  to 

ft.  to 
small 

per  sq.  cm. 
per  cm.  to 

per  in.  to 
small 

per  1  aeg.  cent,  per 
cm.  thick,  to  B.t.u. 

fahr.  per  in.  thick 
to  small  calories 

B.t.u. 

calories 

B.t.u.  per 

calories  per 

per  hr.  per  sq.  ft. 

per  sec.  per  sq.  cm. 

P1tsq. 

per  sq. 
cm. 

sq.  ft. 
per  in. 

sq.  cm. 
per  cm. 

per  1  deg.  fahr. 
per  in.  thick 

per  1  deg.  cent, 
per  cm.  thick 

1 

3.687 

0.2712 

1.451 

0.6892 

2.903X103 

0.033445 

2 

7.374 

0.5424 

2.902 

1.378 

5.806X103 

0.036890 

3 

11.06 

0.8136 

4.353 

2.068 

8.709X103 

0.02I034 

4 

14.75 

1.085 

5.804 

2.757 

11.61  X1Q3 

0.02I378 

5 

18.44 

1.356 

7.255 

3.446 

14.52  X10« 

0.021722 

6 

22.12 

1.627 

8.706 

4.135 

17.42  X1Q3 

0.022067 

7 

25.81 

1.898 

10.16 

4.824 

20.32  X1Q3 

0.022412 

8 

29.50 

2.170 

11.61 

5.514 

23.22  X103 

0.022756 

9 

33.18 

2.441 

13.06 

6.203 

26.13  XlO3 

0.023100 

NOTE.     1  gram-calorie  per  sq.  cm.  =  3.687  B.t.u.  per  sq.  ft. 

1  gram-calorie  per  sq.  cm.  per  cm.  =  1.451  B.t.u.  per  sq.  ft.  per  in. 

1  gram-calorie  per  sec.  per  sq.  cm.  for  a  temp.  grad.  of   1  deg.  cent,  per   cm. 

=  360  kilogram-calories  per  hour  per  sq.  m.  for  a  temp.  grad.  of  1  deg.  cent,  per  m. 

=  2.903  X  103  B.t.u.  per  hour  per  sq.  ft.  for  a  temp.  grad.  of  1  deg.  fahr.  per  in. 


Values  of  Foreign  Coins 

(Legal  standards:    (G)  =  gold;  (S)  =  silver) 


Country 

Monetary 
unit 

Value 
in 
terms 
of  U.  S. 
money 

Country 

Monetary 
unit 

Value 
in 
terms 
of  U.  S. 
money 

Argentina  (G)  

Austria-Hungary  (GO 
Belgium  (G  and  <S) 

Peso  

Crown  
Franc 

CO.  9647 

0.2026 
0  1929 

Great  Britain  (G)  .  .  .  . 

Greece  (G  and  S)  
Haiti  (G) 

Pound  ster- 
ling. 
Drachma.  .  . 
Gourde 

$4.8665 

0.1929 
0  9647 

Bolivia  (G) 

0  3893 

India  (British)  (G) 

0  3244 

Brazil  (G)   .    .  . 

Milreis. 

0  5463 

Italy  (G  and  <S)  

Lira  .  . 

0  1929 

British  colonies  in  .  . 

Pound  ster- 

Japan (G)  

Yen  

0  4984 

Australasia       and 

ling. 

4  8665 

Liberia  (G)     

Dollar  

1  0000 

Africa  (GO 

Mexico  (G) 

Peso 

0  4984 

Canada  (G)  

Dollar....... 

1  0000 

Netherlands  (G)  .... 

Florin  

0  4019 

Central      American 

Norway  (G) 

Crown 

0  2679 

States: 

Panama  (G)  

Balboa  

1  0000 

Coeta  Rica  (GO 

0  4653 

Persia  (G  and  <S) 

Kran  

Variable 

British    Honduras 
(G  or  iS) 

Dollar  

1.0000 

Peru  (G)  
Philippine  Islands  (G) 

Libra  
Peso     

4  8665 
0  5000 

Guatemala  (S)..  .  . 

Peso  

0  4446 

Portugal  (G)  

Escudo  

1  0805 

Honduras  (S)  

Peso  .  . 

0  4446 

Roumania  (G)  

Leu  

0  1929 

Salvador(jS) 

Peso 

0  4446 

Russia  (G) 

Ruble 

0  5145 

Nicaragua  (S) 

1  0000 

Dollar 

1  0000 

Chile  (G)     ... 

Peso 

0  3649 

Servia  (G) 

Dinar.  .. 

0  1929 

China  (S) 

Yuan 

0  4777 

Siam  (G) 

Tical 

0  3708 

Colombia  (G)  

Pound  

4.8665 
0  2680 

Spain  (G  and  <S)  
Straits  Settlement  (G) 

Peseta  
Dollar 

0.1929 
0  5677 

Ecuador  (G)  . 

Sucre 

0  4866 

Sweden  (G)  

Crown  

0  2679 

Egypt  (G)  

Pound  

4  9429 

Switzerland  (CO  

Franc  

0.1929 

Finland  (G) 

Markka 

0  1929 

Turkey  (G)   

Piaster  

0  0439 

France  (G  or  S)  .  .  . 
German  Empire  (G) 

Franc  
Mark  

0.1929 
0  2381 

Uruguay  (G)  
Venezuela  (G)  

Peso  
Bolivar  

1.0340 
0.1929 

TIME  83 

TIME 

Kinds  of  Time.  Three  kinds  of  time  are  recognized  by  astronomers,  viz., 
sidereal,  apparent  solar,  and  mean  solar  time.  The  sidereal  day  is  the  inter- 
val between  two  consecutive  transits  of  some  fixed  celestial  object  across  any 
given  meridian,  or  it  is  the  interval  required  by  the  earth  to  make  one  com- 
plete revolution  on  its  axis.  This  interval  is  constant  but  it  is  inconvenient 
as  a  time  unit  because  the  noon  of  the  sidereal  day  occurs  at  all  hours  of  the 
day  and  night.  The  apparent  solar  day  is  the  interval  between  two  con- 
secutive transits  of  the  sun  across  any  given  meridian.  On  account  of  the 
variable  distance  between  the  sun  and  earth,  the  variable  speed  of  the 
earth  in  its  orbit,  the  effect  of  the  moon,  etc.,  this  interval  is  not  constant 
and  consequently  cannot  be  kept  by  any  simple  mechanism,  such  as 
clocks  or  watches.  To  overcome  the  objection  noted  above,  the  mean 
solar  day  was  devised.  The  mean  solar  day  is  the  length  of  the  average 
apparent  solar  day.  Like  the  sidereal  day  it  is  constant,  and  like  the  apparent 
solar  day  its  noon  always  occurs  at  approximately  the  same  time  of  day.  The 
astronomical  day  begins  at  mean  solar  noon  and  the  hours  run  from  one 
to  twenty-four,  while  the  civil  day  (mean  solar)  begins  12  hours  earlier,  at 
midnight,  and' the  hours  run  from  one  to  twelve,  and  then  repeat  from  noon  to 
midnight. 

The  Year.  There  are  three  different  kinds  of  year  used,  the  sidereal,  the 
tropical,  and  the  anomalistic.  The  sidereal  year  is  the  time  taken  by  the 
earth  to  complete  one  revolution  around  the  sun  from  a  given  star  to  the  same 
star  again.  Its  length  is  365  days,  6  hours,  9  minutes,  and  9  seconds.  The 
tropical  year  is  the  time  included  between  two  successive  passages  of  the 
vernal  equinox  by  the  sun,  and  since  the  equinox  moves  westward  50. "2  of 
arc  a  year,  the  tropical  year  is  shorter  by  20'23."5  in  time  than  the  sidereal 
year.  As  the  seasons  depend  upon  the  earth's  position  with  respect  to  the 
equinox,  the  tropical  year  is  the  year  of  civil  reckoning.  The  anomalistic 
year  is  the  interval  between  two  successive  passages  of  the  perihelion,  namely, 
the  time  of  the  earth's  nearest  approach  to  the  sun.  The  anomalistic  year  is 
only  used  in  special  calculations  in  astronomy. 

The  Calendar.  The  month  depended  originally  upon  the  changes  of  the 
moon.  The  Mohammedan  nations  still  use  a  lunar  calendar  with  years  of 
twelve  lunar  months,  which  alternately  contain  355  and  356  days.  Accord- 
ing to  their  method  of  reckoning  the  same  month  falls  in  different  seasons,  and 
their  calendars  gain  1  year  on  ours  every  33  years.  The  Julian  Calendar 
(established  45  B.  C.)  discards  all  consideration  of  the  moon  and  adopts 
36514  days  as  the  true  length  of  the  year.  It  is  still  used  in  Russia  and 
generally  by  the  Greek  Church.  Gregorian  Calendar:  The  true  length 
of  the  tropical  year  is  365  days,  5  hr.,  48  min.,  45.5  sec.,  a  difference  of  11 
min.,  14.5  sec.  by  which  the  Julian  year  is  too  long.  This  amounts  to  a  little 
more  than  3  days  in  400  years.  To  correct  for  this,  those  century  years  are 
made  leap  years  which  are  divisible  by  400  without  remainder. 

Standard  Time.  Prior  to  1883  each  city  of  the  U.  S.  had  its  own  time, 
which  was  determined  by  the  time  of  passage  of  the  sun  across  the  local  merid- 
ian. A  system  of  standard  time  is  used  at  present,  according  to  which  the 
United  States,  which  extends  from  65  deg.  to  125  deg.  West  longitude,  is  divided 
into  four  sections,  each  of  15  deg.  of  longitude.  The  first  or  eastern  section  in- 
cludes all  territory  between  the  Atlantic  coast  and  an  irregular  line  drawn  from 
Detroit,  Mich.,  through  Pittsburg  to  Charleston,  S.  C.,  its  most  southern 
point.  The  time  of  this  section  is  that  of  the  75-deg.  meridian,  which  is  5 


84 


WEIGHTS  AND  MEASURES 


.„ 


hr.  slower  than  Greenwich  time.  The  second  (central)  section  includes 
territory  between  the  line  mentioned,  and  an  irregular  line  drawn  from  Bis- 
marck, N.  D.,  to  the  mouth  of  the  Rio  Grande.  The  third  (mountain)  sec- 
tion includes  all  territory  between  the  last-named  line  and  a  line  which  passes 
through  the  western  part  of  Idaho,  Utah  and  Arizona.  The  fourth  (Pacific) 
section  covers  the  rest  of  the  country  to  the  Pacific  Ocean.  Standard  time 
is  uniform  in  each  of  these  sections,  but  the  time  in  one  section  differs  by  ex- 
actly 1  hr.  from  the  section  next  to  it.  In  cities  situated  on  the  border  line 
of  two  sections,  as,  say,  Pittsburg  and  Atlanta,  the  standard  times  of  both  sec- 
tions are  used,  and  in  such  cities  when  the  time  is  given,  it  should  be  specified 
as  eastern,  central,  etc.  The  system  of  standard  time  has  been  adopted  in 
almost  all  civilized  countries.  All  continental  Europe,  except  Russia,  uses 
a  time  1  hr.  faster  than  that  of  Greenwich;  in  Japan  and  Australia  the 
time  is  9  hr.  faster. 

TERRESTRIAL  GRAVITY 

By  standard  gravity  is  meant  any  locality  where  g0  =  980.665  cm.  per 
sec.  per  sec.,  or  32.1740  ft.  per  sec.  per  sec.  This  value,  QO,  is  assumed  to  be 
the  value  of  g  at  sea  level  and  latitude  45  deg. 

Acceleration  of  Gravity 

(U.  S.  Coast  and  Geodetic  Survey,  1912) 


Latitude, 

£ 

r 

Latitude, 

/ 

deg. 

Cm./sec.2 

Ft./sec.2 

0/ffo 

deg. 

Cm./sec.2 

Ft./sec.2 

Q/ffo 

0 
10 
20 
30 
40 

978.0 
978.2 
978.6 
979.3 
980.2 

32.088 
32.093 
32.  08 
32.130 
32.158 

0.9973 
0.9975 
0.9979 
0.9986 
0.9995 

50 
60 
70 
80 
90 

981.1 
981.9 
982.6 
.983.1 
983.2 

32.187 
32.215 
32.238 
32.253 
32.258 

.0004 
.0013 
.0020 
0024 
0026 

Correction  for  altitude  above  sea  level:  —  0.3  cm.  per  sec.2  for  each  1000  meters; 
-  0.003  ft.  per  sec.2  for  each  1000  feet. 

SPECIFIC  GRAVITY  AND  DENSITY 

The  specific  gravity  of  a  solid  or  liquid  is  the  ratio  of  the  mass  of  the 
body  to  the  mass  of  an  equal  volume  of  water  at  some  standard  temperature. 
At  the  present  time  a  temperature  of  4  deg.  cent.  (39  deg.  fahr.)  is  commonly 
used  by  physicists,  but  the  engineer  uses  60  deg.  fahr.  The  specific  gravity 
of  gases  is  usually  expressed  in  terms  of  hydrogen  or  air. 

The  density  of  a  body  is  its  mass  per  unit  volume.  If  the  gram  is  used  as 
the  unit  of  mass  and  the  milliliter  as  the  unit  of  volume,  the  figures  represent- 
ing the  density  are  the  same  as  the  specific  gravity  of  the  body  referred  to 
water  at  4  deg.  cent,  as  unity.  The  customary  unit  is  pounds  per  cu.  ft. 

The  specific  gravity  of  liquids  is  usually  measured  by  means  of  an  hydrom- 
eter (see  p.  254).*  Special  arbitrary  hydrometer  scales  are  used  in  various 
trades  and  industries.  The  most  common  of  these  are  the  Baum6,  Twaddell 
and  Beck.  Twaddell's  hydrometer  is  used  for  liquids  heavier  than  water. 
The  number  of  degrees,  N,  which  it  indicates  may  be  converted  to  specific 
gravities,  G,  by  the  formula  G  =  (5N  +  1000) /1000.  The  formula  for 
the  Beck  hydrometer  is  G  =  170/(170  ±  N);  for  the  Brix  hydrometer  G  = 
400/(400  ±  N).  In  both  of  these  the  +  sign  is  to  be  used  for  liquids  lighter 
than  water,  the  —  sign  for  heavier  liquids.  For  the  salinometer  (salometer), 
see  p.  1734.  *  The  specific  gravities  corresponding  to  the  indications  of  the 
Baum6  hydrometer  are  given  in  the  following  tables. 
•See  Marka'  MECHANICAL  ENGINEERS'  HANDBOOK. 


SPECIFIC  GRAVITY  AND  DENSITY 


85 


60° 


Specific  Gravities  at  —  Fahr.  Corresponding  to  Degrees  Baume* 
for  Liquids  Lighter  than  Water 


60° 
Calculated  from  the  formula,  specific  gravity  —5  fahr.  = 


140 


130  +  Deg. 


Degrees 
Baume 

Specific 
gravity 

Degrees 
Baume 

II 
11 

02  M 

|| 

$$ 
Qpq 

Specific 
gravity 

s*> 

Si 

aa 

Specific 
gravity 

Degrees 
Baumfi 

Specific 
gravity 

Degrees 
Baum6 

Specific 
gravity 

10 

1.0000 

25 

0.9032 

40 

0.8235 

55 

0.7568 

70 

0.7000 

85 

0.6512 

11 

0.9929 

26 

0.8974 

41 

0.8187 

56 

0.7527 

71 

0.6965 

86 

0.6482 

12 

0.9859 

27 

0.8917 

42 

0.8140 

57 

0.7487 

72 

0.6931 

87 

0.6452 

13 

0.9790 

28 

0.8861 

43 

0.8092 

58 

0.7447 

73 

0.6897 

88 

0.6422 

14 

0.9722 

29 

0.8805 

44 

0  8046 

59 

0.7407 

74 

0.6863 

89 

0.6393 

15 

0.9655 

30 

0.8750 

45 

0.8000 

60 

0.7368 

75 

0.6829 

90 

0.6364 

16 

0.9589 

31 

0.8696 

46 

0.7955 

61 

0.7330 

76 

0.6796 

91 

0.6335 

17 

0.9524 

32 

0.8642 

47 

0.7910 

62 

0.7292 

77 

0.6763 

92 

0.6306 

18 

0.9459 

33 

0.8589 

48 

0.7865 

63 

0.7254 

78 

0.6731 

93 

0.6278 

19 

0.9396 

34 

0.8537 

49 

0.7821 

64 

0.7216 

79 

0.6699 

94 

0.6250 

20 

0.9333 

"  35 

0.8485 

50 

0.7778 

65 

0.7179 

80 

0.6667 

95 

0.6222 

21 

0.9272 

36 

0.8434 

51 

0.7735 

66 

0.7143 

81 

0.6635 

96 

0.6195 

22 

0.9211 

37 

0.8383 

52 

0.7692 

67 

0.7107 

82 

0.6604 

97 

0.6167 

23 

0.9150 

38 

0.8333 

53 

0.7650 

68 

0.7071 

83 

0.6573 

98 

0.6140 

24 

0.9091 

39 

0.8284 

54 

0.7609 

69 

0.7035 

84 

0.6542 

99 

0.6114 

100 

0.6087 

60° 

Specific    Gravities  at  —  Pahr.  Corresponding  to  Degrees  Baume 


for  Liquids  Heavier  than  Water 

Calculated  from  the  formula,  specific  gravity         fahr.  = 


Degrees 
Baum6 

Specific 
gravity 

Degrees 
Baume 

Specific 
gravity 

Degrees 
Baum6 

°  >> 

11 

02  00 

ii 

Specific 
gravity 

Degrees 
Baume 

1 

Degrees 
Baume 

Specific 
gravity 

0 

.0000 

12 

.0902 

24 

.1983 

36 

.3303 

48 

.4948 

60 

.7059 

1 

.0069 

13 

.0985 

25 

.2083 

37 

.3426 

49 

.5104 

61 

.7262 

2 

.0140 

14 

.1069 

26 

.2185 

38 

.3551 

50 

.5263 

62 

.7470 

3 

.0211 

15 

.1154 

27 

.2288 

39 

.3679 

51 

.5426 

63 

.7683 

4 

.0284 

16 

.1240 

28 

.2393 

40 

.3810 

52 

.5591 

64 

.7901 

5 

.0357 

17 

.1328 

29 

.2500 

41 

.3942 

53 

.5761 

65 

.8125 

6 

.0432 

18 

.1417 

30 

.2609 

42 

.4078 

54 

.5934 

66 

.8354 

7 

.0507 

19 

.1508 

31 

.2719 

43 

.4216 

55 

.6111 

67 

.8590 

8 

.0584 

20 

.1600 

32 

.2832 

44 

.4356 

56 

.6292 

^68 

.8831 

9 

.0662 

21 

.1694 

33 

.2946 

45 

.4500 

57 

.6477 

69 

.9079 

10 

.0741 

22 

.1789 

34 

.3063 

46 

.4646 

58 

.6667 

70 

.9333 

11 

.0821 

23 

.1885 

35 

.3182 

47 

.4796 

59 

.6860 

Mohs's  Scale  of  Hardness 

1.  Talc.    2.  Gypsum.    3.  Calc  spar.    4.  Fluorspar.     5.  Apatite. 
6.  Feldspar.    7.  Quartz.     8.  Topaz.    9.  Sapphire.     10.  Diamond. 


SECTION  2 
MATHEMATICS 


BY 
EDWARD  V.  HUNTINGTON,  Ph.  D. 

ASSOCIATE   PROFESSOR  OF  MATHEMATICS,  HARVARD   UNIVERSITY,  FELLOW   AM.  ACAD. 
ARTS  AND  SCIENCES 


CONTENTS 


ARITHMETIC 

PAGE 

Numerical  Computation 88 

Logarithms 91 

The  Slide  Rule 94 

Computing  Machines 97 

Financial  Arithmetic 98 

GEOMETRY  AND  MENSURATION 

Geometrical  Theorems 99 

Geometrical  Constructions 101 

Lengths  and  Areas  of  Plane  Figures.  105 

Surfaces  and  Volumes  of  Solids 107 

ALGEBRA 

Formal  Algebra 112 

Solution  of   Equations  in  One  Un- 
known Quantity 116 

Solution  of  Simultaneous  Equations  119 

Determinants 123 

Imaginary   or    Complex    Quantities  124 

TRIGONOMETRY 

Formal  Trigonometry 128 

Solution  of  Plane  Triangles 132 

Solution  of  Spherical  Triangles 134 

Hyperbolic  Functions 135 

ANALYTICAL  GEOMETRY 

The  Point  and  the  Straight  Line.  ...  136 

The  Circle 137 


PAOE 

The  Parabola 138 

The  Ellipse 140 

The  Hyperbola 144 

The  Catenary 147 

Other  Useful  Curves 151 

DIFFERENTIAL  AND  INTEGRAL 
CALCULUS 

Derivatives  and  Differentials 1 57 

Maxima  and  Minima 159 

Expansion  in  Series 160 

Indeterminate  Forms 163 

Curvature 163 

Table  of  Indefinite  Integrals 164 

Definite  Integrals 169 

Differential  Equations 171 

GRAPHICAL  REPRESENTATION  OF 
FUNCTIONS 

173 
174 
176 
177 
178 
182 


Equations  Involving  Two  Variables 
Equations  for  Empirical  Curves . 
Logarithmic  Cross-section  Paper . 

Semi-logarithmic  Paper 

Equations  Involving  Three  Variables 
Equations  Involving  Four  Variables 


VECTOR  ANALYSIS 

Vector  Analysis 185 


COPYRIGHT,  1916,  BY  EDWARD  V.  HUNTINGTON 


MATHEMATICS 

BY 
EDWARD  V.  HUNTINGTON 


ARITHMETIC 

NUMERICAL  COMPUTATION 

Number  of  Significant  Figures.  In  any  engineering  computation,  the 
data  are  ordinarily  the  results  of  measurement,  and  are  correct  o'nly  to  a 
limited  number  of  significant  figures.  Each  of  the  numbers  3.840  and 
0.003840  is  said  to  be  given  "correct  to  four  figures;"  the  true' value  lies  in 
the  first  case  between  3.8395  and  3.8405;  in  the  second  case,  between  0.0038395 
and  0.0038405.  The  absolute  error  is  less  than  0.001  in  the  first  case,  and 
less  than  0.000001  in  the  second;  but  the  relative  error  is  the  same  in  both 
cases,  namely,  an  error  of  less  than  "one  part  in  3840." 

If  a  number  is  written  as  384000,  the  reader  is  left  in  doubt  whether  the  number  of 
correct  significant  figures  is  3,  4,  5,  or  6.  This  doubt  can  be  removed  by  writing  the 
number  as  3.84  X  10»  or  3.840  X  10s  or  3.8400  X  10*  or  3.84000  X  105. 

In  any  numerical  computation,  the  possible  or  desirable  degree  of  accuracy 
should  be  decided  on  and  the  computation  should  then  be  so  arranged  that 
the  required  number  of  significant  figures,  and  no  more,  is  secured.  Carry- 
ing out  the  work  to  a  larger  number  of  places  than  is  justified  by  the  data,  is 
to  be  avoided,  (1)  because  the  form  of  the  results  leads  to  an  erroneous  impres- 
sion of  their  accuracy,  and  (2)  because  time  and  labor  are  wasted  in  super- 
fluous computation.  The  labor  of  working  with  six-place  tables  is  nearly 
three  times  as  great  as  that  with  four-place  tables.  In  computations  involv- 
ing several  steps,  it  is  desirable  to  retain  one  extra  figure  until  just  before  the 
final  result  is  reached,  in  order  to  protect  the  last  figure  against  the  possible 
cumulative  effect  of  small  tabular  errors.  In  discarding  superfluous 
figures,  if  the  first  discarded  figure  is  5  or  more,  increase  the  preceding 
figure  by  1.  Thus,  3.14159,  written 'correct  to  four  figures,  is  3.142;  correct 
to  three  figures,  3.14.  Again,  6.1297,  correct  to  four  figures,  is  6.130. 

Addition.     In  adding  numbers,  note  that  a  doubtful  final  0 . 2056x 

figure  in  any  one  number  will  render  doubtful  the  whole  col-  2 . 572xx 

umn  in  which  that  figure  lies;  hence  all  figures  to  the  right  of  14.25xxx 

that  column  are  superfluous,  and  contribute  nothing  to  the  576.1xxxx 

accuracy  of  the  result. 

Subtraction.     The    "Austrian"    or    "shop"    method  is  593.1 

recommended.     The  mental  process  is  as  follows,  the  figures  here  printed  in 
boldface  type  being  the  only  ones  written  down: 

[3  plus  how  many  is  12?]  3  plus  9  is  12;  1  to  carry.  14752 

[7  plus  how  many  is  15?]  7  plus  8  is  15;  1  to  carry.  ,J^5§, 

5  plus  2  is  7.     8  plus  6  is  14.  6289 

88 


NUMERICAL  COMPUTATION  89 

This  method  is  especially  useful  when  it  is  desired  to  subtract  from  a  given 
number  the  sum  of  several  other  numbers. 

7  plus  1  is  8;  plus  5  is  13;  plus  9  is  22;  2  to  carry.  14752 

5  plus  0  is  5;  plus  2  is  7;  plus  8  is  15;  1  to  carry.  3125~| 

3  plus  1  is  4;  plus  1  is  5;  plus  2  is  7.  101 

5  plus  3  is  8;  plus  6  is  14.  _5237-J 

6289 

The  use  of  a  wavy  line  to  indicate  subtraction  is  also  recommended,  as  it 
will  minimize  the  danger  of  adding  when  subtraction  is  intended. 

Multiplication.     In    long    examples    in  multiplication,  4956 

the  arrangement  of  work  here  illustrated  is  recommended,  8372 


since  it   facilitates   the   abbreviation   of   the   work  by  the  39648 

omission,  in  practice,  of  all  the  figures  on  the  right  of  the  1486  8 

vertical  line.  346  92 

The  position  of  the  decimal  point  should  be  determined  9  912 


by  reference'to  the  first,  or  left-hand,  figures  of  the  numbers,  41492|xxx 

rather  than  by  "pointing  off"  so-and-so  many  places  from 
the  right-hand  end.     For  the  right-hand  figures  of  a  number  are  the  least 
important  ones,  and  in  many  cases  are  entirely  unknown  (especially  when 
the  slide  rule  or  a  computing  machine  is   used).     The  mental  process  for 
determining  the  decimal  point  is  as  follows: 

(a)  If  the  multiplier  is  a  number  like  3.1416,  with  only  one  figure  preceding 
the  decimal  point,  think  of  this  number  as  "a  little  over  3;"  then  the  product 
must  be  "a  little  over  three  times  the  number  which  is  being  multiplied;" 
and  this  gives  the  position  of  the  decimal  point  at  once,  by  inspection. 

(6)  If  the  multiplier  is  a  number  like  3141.6  [or  0.000  003  141  6],  think  of 
this  number  as  "about  3,  with  the  point  moved  three  places  to  the  right" 
[or  "about  3,  with  the  point  moved  six  places  to  the  left"];  then  think  what 
the  answer  would  be  if  the  multiplier  were  simply  "about  3,"  and  shift  the 
decimal  point  accordingly. 

Multiplication  Tables.  Crelle's  large  volume  (Berlin,  G.  Reimer)  gives  the  product 
of  every  three-figure  number  by  every  three-figure  number;  Peters's  (Berlin,  G.  Reimer), 
of  every  four-figure  number  by  every  two-figure  number.  The  smaller  table  of  H. 
Zimmermann  (Berlin,  Wm.  Ernst)  gives  the  product  of  every  three-figure  number  by 
every  two-figure  number. 

Division.  In  long  division,  where  the  numbers  are  given  23026)31416(1 
only  approximately,  the  work  can  be  much  abbreviated  with-  23026 

out  loss  of  accuracy  by  "cutting  off"  one  figure  of  the  divisor      2303)   8390(3 
at  each  step,  instead  of  "bringing  down"  a  doubtful  zero  in  6909 

the  dividend.     Thus,  3.1416  4-  2.3026  =  1.3644.  „ 

To  determine  the  position  of  the  decimal  point  in  a         du; 
problem  of  fractional  division,  shift  the  point  (mentally)  in 
both   numerator  and   denominator  (the    same   number  of          23)      101(4 
places  in  each)   until  the  denominator  is   a  number  in  the 
"standard  form,  "  that  is,  a  number  with  only  one  figure  pre-  2)          9(4 

ceding  the  decimal  point.     (This  will  not  change  the  value 
of  the  fraction.)     Then  estimate  the  approximate  magnitude  of  the  quotient 
by  inspection.     Thus: 

0.2718       0.000  2718 

"about  0.000  09"   =  0.000  08652; 


3141.6  3.1416 

31.416        31  416. 


0.002718        2.718 


"about  10  000"  =11  558. 


90  ARITHMETIC 

Reciprocals.  The  reciprocal  of  N  is  1  /N.  Instead  of  dividing  by  a  long 
number  N,  it  is  often  better  to  multiply  by  the  reciprocal  of  N.  The  table 
of  reciprocals  on  pp.  24-27  gives  the  reciprocal  of  any  number,  correct  to 
four  figures.  Barlow's  Table  (Spoil  &  Chamberlain,  New  York)  gives  the 
reciprocal  of  every  four-figure  number  correct  to  seven  figures  (but  with- 
out facilities  for  interpolation).  The  reciprocals  of  numbers  having 
more  than  four  figures  are  best  found  by  the  use  of  a  large  table  of 
logarithms. 

Reciprocals  of  I  +  x  when  x  is  Small. 
1/(1  +  x)  =  1  -  x  +  [error  <  x2,  if  x  is  between  0  and  1], 

=  1  —  x  +  xz  —  [error  <  x3,  if  x  is  between  0  and  1]. 
1/(1  —  x)  =  1  +  x  +  [error  <  x2  +  2z3,  if  x  is  between  0  and  }*], 

=  1  +  x  +  x2  +  [error  <  x3  +  2z4,  if  x  is  between  0  and  #]. 

NOTE.  l/(o  ±  6)  =  (l/a)[l/(l  ±  «)],  where  x  =  6/a. 

Notation  by  Powers  of  10.  All  questions  concerning  the  position  of  the 
decimal  point  are  readily  answered  if  each  number  is  expressed  in  the  "stand- 
ard form,"  that  is,  as  the  product  of  two  factors,  one  of  which  is  a  number 
with  only  one  figure  preceding  the  decimal  point,  while  the  other  is  a  positive 
or  negative  power  of  10.  Thus,  3.1416  X  103  means  3.1416  with  the  point 
moved  three  places  to  the  right,  that  is,  3141.6.  Again,  3.1416  X  10~6  means 
3.1416  with  the  point  moved  six  places  to  the  left,  that  is,  0.000  003  1416. 
This  notation  by  powers  of  10  should  always  be  used  in  dealing  with  very 
large  or  very  small  numbers.  Among  electrical  engineers  its  use  is  very 
general,  even  for  numbers  of  moderate  size. 

Square  Root,  (a)  If  four  figures  of  the  root  are  sufficient,  take  the 
answer  directly  from  the  table  of  square  roots,  pp.  12-15.  (6)  To  obtain  a 
root  of  six  or  seven  figures  from  the  table,  use  the  formula:  VJV  =  a  + 
[(N  —  ,a2)/2a]  (approx.),  where  a  is  the  nearest  value  of  v~N  obtainable 
from  the  table,  with  three  or  four  ciphers  annexed.  Here  a2  must  be  found 
exactly,  by  direct  multiplication,  so  that  at  least  three  significant  figures 
of  the  difference  N  —  a2  shall  be  known  correctly;  but  this  done,  the  division 
of  N  —  a2  by  2a  should  be  carried  to  only  three  figures  (logarithms  or  slide 
rule  may  be  used). 

NOTE.  The  simplest  way  to  obtain  any  root  of  a  seven-figure  number  correct  to 
seven  figures  is  to  use  a  seven-place  table  of  logarithms,  if  such  a  table  is  at  hand. 

Square  Roots  of  1  ±  x  when  x  is  Small. 

(1  +  *)**   =  1  +  \hx  -  [error  less  than  ftx2  if  0  <  x  <  1] 
=  1  +  Kx  -  Hxz  +  [error  <  H«  *3  if  0  <  x  <  1] 

(1  -  x)W   =  1  -  MX  -  [error  <  #ca+  Mo*3  if  0  <  x  <  J4] 

=  1  -  )6x  -  ^z2 -[error  <  Ho*3  +  Hex4  if  0  <  x  <  ft] 

NOTE.     Va  +  b  =  Va  (1  +  x)^t  where  x  =  6/a. 

Cube  Root,  (a)  If  four  figures  of  the  root  are  sufficient,  take  the  answer 
directly  from  the  table  of  cube  roots,  pp.  16-21.  (6)  To  obtain  a  root  of 
six  or  seven  figures  from  the  table,  use  the  formula:  %/N  =  a  +  [(N  —  o3)/3a2] 
(approx.),  where  a  is  the  nearest  value  of  $/N  obtainable  from  the  table,  with 
three  or  four  ciphers  annexed.  Here  a3  must  be  found  correct  to  seven  or 
eight  figures,  by  direct  multiplication,  so  that  at  least  three  significant  figures 
of  the  difference  N  —  a3  shall  be  known;  but  this  done,  the  division  of  N  —  a8 
by  3o2  should  be  carried  to  only  three  or  four  figures  (logarithms  or  the  slide 
rule  may  be  used). 


LOGARITHMS  91 

NOTE.  The  simplest  way  to  obtain  any  root  of  a  seven-figure  number  correct  to 
•seven  figures  is  to  use  a  seven-place  table  of  logarithms,  if  such  a  table  is  at  hand. 

Cube  Roots  of  1+x  when  x  is  Small. 

(1  +  z)H    =  1  +  MX  -  [error  <  H*2  if  0  <  a:  <  1], 

=  1  +  %x  -  Kx*  +  [error  <  Me*3  if  0  <  *  <  1], 
(1  -  x)%    =  1  -  %x  -  [error  <  %x*  +  Mo*3  if  0  <  x  <  H], 

=  1  -  MX  -Jte2  -  [error  <  Me*3  +  Ms*4  if  0  <  x  <  ft]. 
NOTE.     3/a  +  6  =  ^/a(l  +  z)**,  where  x  =  b/a. 

LOGARITHMS 

Tables  of  Logarithms.  The  use  of  a  table  of  logarithms  greatly  reduces 
the  labor  of  multiplication,  division,  raising  to  powers,  and  extracting  roots. 
The  table  on  pp.  42-43  is  carried  out  to  four  significant  figures,  and  the  follow- 
ing explanations  should  be  sufficient  to  permit  the  use  of  the  table  readily, 
even  by  one  without  previous  experience.  For  algebraic  theory,  see  p.  113. 

If  more  than  four-figure  accuracy  is  required,  recourse  must  be  had  to  a  larger  table. 
Five-place  tables  are  available  in  great  variety;  the  Macmillan  Tables,  1913,  are  perhaps 
as  convenient  as  any.  If  more  than  five  figures  are  required,  use  Bremiker's  six-place 
table,  or  proceed  at  once  to  a  seven-place  table:  Schron  (Vieweg  und  Sohn,  Braun- 
schweig); Bruhns;  Vega-Bremiker.  If  extreme  accuracy  is  required,  use  the  eight-place* 
table  by  Bauschinger  and  Peters  (Engelmann,  Leipzig).  Logarithmic  paper,  see  p.  176. 

To  Find  the  Logarithm  of  Any  Given  (Positive)  Number. 

(a)  WHEN  THE  GIVEN  NUMBER  is  BETWEEN  1  AND  10. 

An  inspection  of  the  table  on  pp.  42-43  shows  that  as  the  number  increases 
from  1  to  9.99.  .  .  the  logarithm  of  that  number  increases  continuously  from 
0  to  0.999.  .  .  For  example,  log  2.97  =  0.4728;  log  2.98  =  0.4742. 

If  the  given  number  contains  four  significant  figures,  it  is  necessary  to  inter- 
polate between  the  tabulated  values,  as  follows: 

To  find  log  2.973,  notice  that  this  number  is  fio  of  the  way  from  2.97  to  2.98; 
hence  its  logarithm  will  be  (approximately)  Mo  of  the  way  from  0.4728  to  0.4742.  The 
difference  here  is  14  units,  and  iHo  of  this  difference  is  4  (to  the  nearest  unit);  hence, 
by  adding  this  4  to  4728,  log  2.973  =  0.4732.  This  process  of  interpolating  should 
be  performed  mentally;  the  step  of  finding  the  tabular  difference  will  be  facilitated  by 
a  glance  at  the  last  column  on  the  right,  which  gives,  for  each  line  of  the  table,  the 
average  of  the  differences  along  that  line. 

Again,  to  find  log  4.098:  From  table,  log  4.09  =  0.6117;  adding  9io  of  the  difference 
(11),  or  about  9,  gives:  log  4.098  =  0.6126.  Or  better,  since  91o  °f  the  way  forward 
is  equal  to  Y\Q  of  the  way  back,  find  in  table  log  4.10  •=  0.6128,  and  subtract  Y\Q  of  11, 
or  2,  giving  log.  4.098  =  0.6126.  It  should  be  noted  that  any  interpolated  value  may 
be  in  error  by  1  in  the  last  place. 

If  the  given  number  contains  more  than  four  significant  figures,  it  should 
be  cut  down  to  four  figures  (see  p.  88),  since  the  later  figures  will  not  affect 
the  result  in  four-place  computations. 

(6)  WHEN  THE  GIVEN  NUMBER  is  LESS  THAN  1  OR  MORE  THAN  10,  it  is  simply 
necessary  to  notice  that  every  such  number  can  be  regarded  as  obtainable 
from  some  number  between  1  and  10  by  merely  shifting  the  decimal  point 
(see  p.  90) ;  and  that  according  to  the  rule  at  the  foot  of  the  table,  moving 
the  decimal  point  n  places  to  the  right  [or  left]  in  the  number-column  is 
equivalent  to  adding  n  [or  —  n]  to  the  logarithm  in  the  body  of  the  table. 

For  example,  to  find  log  2973.  Here  2973  =  2.973  X  10«  (i.e.,  2.973  with  the 
decimal  point  moved  3  places  to  the  right).  From  the  table,  log  2.973  «=>  0.473i. 
Hence,  log  2973  =  0.4732  +  3,  which  may  be  written  as  3.4732. 


92  ARITHMETIC 

Again,  to  find  log  0.0002973.  Here  0.0002973  =  2.973  X  10~*  (i.e.,  2.973  with  the 
decimal  point  moved  4  places  to  the  left).  From  the  table,  log  2.973  =  0.4732.  Hence, 
log  0.0002973  =  0.4732  -  4.  (This  may  be  written  as  4.4732,  if  desired,  and  is  equal 
of  course,  to  _—  3.5268;  this  latter  form,  however,  is  not  convenient  in  practice.) 

It  is  thus  evident  that  the  logarithm  of  every  positive  number  may  be 
regarded  as  consisting  of  two  parts:  a  decimal  fraction,  which  is  always  posi- 
tive (or  zero) ;  and  a  whole  number,  which  may  be  positive,  negative,  or  zero. 
The  fractional  part  is  called  the  mantissa,  and  is  found  from  the  table ;  the 
whole-number  part  is  called  the  characteristic,  and  is  determined  by 
inspection. 

To  Find  the  Number  Corresponding  to  a  Given  Logarithm. 

(a)  WHEN  THE  GIVEN  LOGARITHM  is  A  POSITIVE  DECIMAL  FRACTION  (CHARAC- 
TERISTIC ZERO),  simply  reverse  the  process  for  finding  the  logarithm  of  a 
number  between  1  and  10. 

For  example,  given  log  N  =  0.4732;  to  find  N.  In  the  body  of  the  table  it  is  seen 
that  0.4732  lies  a  little  beyond  0.4728;  hence  N  must  lie  a  little  beyond  2.97.  By  taking 
differences  it  is  found  that  4728  is  in  fact  #4  of  the  way  from  0.4728  to  the  next 
higher  logarithm;  therefore  N  must  be  y\±  of  the  way  from  2.97  to  the  next  higher 
number.  But  YU  of  1  is  0.3  (to  the  nearest  tenth),  hence  N  =  2.973. 

Again,  given  log  N  =  0.6126;  to  find  N.  Here,  0.6126  is  %i  of  the  way  from  0.6117 
to  the  next  higher  logarithm;  therefore  N  must  be  JH \  of  the  way  from  4.09  to  the  next 
higher  number.  But  %\  of  1  is  0.8  (to  the  nearest  tenth),  hence  N  =  4.098. 

(6)  WHEN  THE  GIVEN  LOGARITHM  HAS  ANY  GIVEN  VALUE  (CHARACTERISTIC 
NOT  ZERO),  proceed  as  follows:  First,  be  sure  the  given  logarithm  is  in  the 
"standard  form,"  that  is,  a  positive  decimal  fraction  (mantissa)  plus  a  posi- 
tive or  negative  whole  number  (characteristic).  For  example,  if  log  N  is 
originally  given  in  the  form  log  N  =  —  3.5268,  tfiis  must  first  be  reduced  to 
the  (equivalent)  form  log  N  =  0.4732  —  4  (or  4.4732),  before  entering  the 
table.  Having  the  logarithm  given  in  the  standard  form,  suppose  for  the 
moment  that  the  characteristic  is  zero,  and  find  in  the  table  the  number 
corresponding  to  the  given  mantissa;  then  move  the  decimal  point  to  the  right 
or  left  according  as  the  value  of  the  characteristic  is  positive  or  negative. 

For  example,  given  log  N  =  0.4732  +  3;  to  find  N.  From  the  table,  the  number 
corresponding  to  0.4732  is  2.973.  The  characteristic  (  +  3)  directs  that  the  decimal 
point  be  moved  3  places  to  the  right;  hence  N  =  2.973  X  103  =  2973. 

Again,  given  log  N  =  0.4732  —  4;  to  find  N.  From  the  table,  the  number  corre- 
sponding to  0.4732  is  2.973.  The  characteristic  (  -  4)  indicates  that  the  decimal 
point  is  to  be  moved  4  places  to  the  left;  hence  N  =  2.973  X  10~«  =  0.0002973. 

The  number  corresponding  to  a  given  logarithm  is  called  its  antiloga- 
rithm.  Thus,  if  log  2973  =  0.4732  +  3,  then  2973  =  antilog  (0.4732  +  3). 

NOTE  1.  In  most  tables  of  logarithms  the  decimal  point  is  omitted,  the  tables  being 
in  fact  not  tables  of  logarithms,  but  tables  of  mantissas.  This  omission  is  of  no  con- 
sequence to  the  experienced  computer,  but  is  often  perplexing  to  one  who  makes  only 
occasional  use  of  such  tables. 

NOTE  2.  Many  computers  prefer  to  write  negative  characteristics  in  the  form  of  some 
positive  number  minus  some  multiple  of  10;  thus,  0.4732  —  4  =  6.4732  —10; 
0.4732  -  13  =  7.4732  -  20;  etc. 

Fundamental  Properties  of  Logarithms.  The  usefulness  of  logarithms 
in  computation  depends  on  the  following  properties: 

(1)  log  (a&)    =  log  a  +  log  6;     (3)  log  (a71)   =  n  log  a; 

(2)  log  (a/6)  =  log  a  -  log  6;     (4)  log  \/a  =  (1/n)  log  a; 

(5)  log  10n  =  n 
It  is  to  be  noted  also  that  log  1  =  0,  log  10  =1,  and  log  (1/n)   =   —log  n. 


LOGARITHMS  93 

To  Multiply  by  Logarithms.  Find  from  the  table  the  log.  of  each  factor, 
and  add;  the  result  will  be  the  log.  of  the  product.  Then  find  the  product 
itself  from  the  table. 

EXAMPLE".     To  find  log  4.098  =  0.6126 

x  -  (4.098)  (0.0002973)  (72.1).  log  0.0002973  -  0.4732  -  4 

Answer:  x  =  8.784  X  lO"'  l°g  72.1  -  0.8579  -f  1 

=  0.08784  log  x  =  1.9437  -  3  -  0.9437  -  2. 

To  Divide  by  Logarithms.  First  Method:  Find  from  the  table  the 
log.  of  the  numerator  and  the  log.  of  the  denominator,  and  subtract  the  second 
from  the  first;  the  result  will  be  the  logarithm  of  the  quotient.  Then  find  the 
quotient  itself  from  the  table. 

4.098  log    4.098  -  0.6126 

EXAMPLE.      To  find  x  =  ^^         ^    ^^  =  ^732^ 

Answer:  x  -  1.378  X  10«  =  13780          log    x  -  0.1394  -f  4 

In  order  to  avoid  negative  mantissas  in  cases  where  a  larger  mantissa 
would  have  to  be  subtracted  from  a  smaller,  modify  the  upper  logarithm  by 
adding  and  subtracting  1. 

0.0291  log  0.0291  -  0.4639  -  2  =  1.4639  -  3 

EXAMPLE.      To  find  x  =  •  _  _ 

Answer:   x  =  4.590  X  10~«  log  3  "*"-  06618^4 

=  0.0004590. 

But  if  the  logarithms  are  written  with  the  characteristics  in  front,  and  the  "shop 
method"  of  subtraction  is  used  (see  p.  ]Og  Q  0291  «=  24639 

88),  then  no    such  special  device  is  here  jog    §3  4  •  ,_,  j  g021 

required.     Thus:  T~ 

log    x  =  4.6618 

To  Divide  by  Logarithms.  Second  Method:  Instead  of  subtracting 
the  log.  of  a  number,  it  is  often  convenient  to  add  the  cologarithm  of  that 
number;  the  colog.  of  N  being  defined  by:  colog  N  =  log  (l/N)  =  —log  N. 

To  find  the  colog.  of  a  number,  write  the  log.  of  the  number  in  the  stand- 
ard form,  and  subtract  it  from  1.0000  —  1,  as  in  the  following  examples: 

1.0000  -  1  1.0000  -  1 

log  69.5  =  0.8420  +  1  log  0.0002973  =  £.4732_-j4 

colog  69.5  =  0.1580  -  2  colog  0.0002973  =  0.5268   +  3 

This  subtraction  should  be  performed  mentally.  Thus,  to  subtract  the  mantissa, 
subtract  each  digit  from  9  until  the  last  non-zero  digit  is  arrived  at,  and  subtract  this 
from  10;  to  subtract  the  characteristic,  follow  the  regular  rule  of  algebra  ("reverse  the 
sign  and  add").  Hence,  if  the  logarithm  itself  is  already  written  down,  or  can  be  read 
off  from  the  table  without  interpolation,  the  cologarithm  can  be  written  down  at  once, 
by  inspection.  The  use  of  cologarithms  is  not  essential  in  logarithmic  computation,  but 
it  often  facilitates  a  compact  arrangement  of  the  work,  especially  in  cases  where  the 
denominator  of  a  fraction  is  itself  the  product  of  two  or  more  factors. 

To  Find  the  nth  Power  of  a  Number  by  Logarithms.  Find  from  the 
table  the  log.  of  the  number,  and  multiply  it  by  n;  the  result  will  be  the 
logarithm  of  the  nth  power  of  that  number.  Then  find  the  power  itself  from 
the  tables. 

EXAMPLE  1.     Find  x  =  (0.0291)s  log  0.0291  =  0.4639  -  2 

Answer:     x  =  2.464  X  10~«  3 

=  0.00002464.  log  x  =  1.3917  -  6  -  0.3917  -  5. 


94  ARITHMETIC 

EXAMPLE  2.     Find    x  =  (0.0291)i'«i          log  0.0291  =  0.4639  -  2  =  -  1.5361 

Answer:     x  =  6.825  X  10~8  1.41 

=  0.006825  15361~ 

61444 

15361 

logs          =         -  2.1659 
=    0.8341  -  3 

To  Find  the  nth  Root  of  a  Number  by  Logarithms.  Find  from  the 
table  the  log.  of  the  number,  and  divide  it  by  n;  the  result  will  be  the  log.  of 
the  nth  root  of  that  number.  Then  find  the  root  itself  from  the  table. 

EXAMPLE.      Find  x  =  ^/4.098  log  4.098  =  0.6126 

Answer:     x  =       1.600  log  x  =  0.2042 

In  order  to  avoid  fractional  characteristics,  if  the  characteristic  is 
not  divisible  by  n,  make  it  so  divisible  by  adding  and  subtracting  a  suitable 
number  before  dividing. 


EXAMPLE.      Find  x  =  VO-0004590.  log  0.0004590  =  0.6618  -  ' 

Answer:     x  =  7.714  X  10-2  3)2.6618  -  6 

=  0.07714  log  x  =  0^8873  -  2 

But  if  the  characteristic  is  positive,  it  is  simpler  to  write  it  in  front  of  the  mantissa, 
and  then  divide  directly. 

THE  SLIDE  RULE 

The  slide  rule  is  an  indispensable  aid  in  all  problems  in  multiplication, 
division,  proportion,  squares,  square  roots,  etc.,  in  which  a  limited  degree 
of  accuracy  is  sufficient.  The  ordinary  10-in.  Mannheim  rule  (see  below) 
costs  $3  to  $4.50  and  gives  three  significant  figures  correctly;  the  20-in. 
rule  ($12.50)  gives  from  three  to  four  figures;  the  Fuller  spiral  rule  ($30) 
or  the  Thacher  cylindrical  rule  ($35)  gives  from  four  to  five  figures.  For 
many  problems  the  slide  rule  gives  results  more  rapidly  than  a  table  of  loga- 
rithms; it  requires,  however,  more  care  in  placing  the  decimal  point  in  the 
answer.  In  all  work  with  the  slide  rule,  the  position  of  the  decimal  point 
should  be  determined  by  inspection  (see  p.  89),  only  the  sequence  of  digits 
being  obtained  from  the  instrument  itself.  Rapidity  in  the  use  of  the  in- 
strument depends  mainly  on  the  skill  with  which  the  eye  can  estimate  the 
values  of  the  various  divisions  on  the  scale;  expertness  in  this  respect  comes 
only  with  practice.  The  following  explanations  should  be  sufficient  to  per- 
mit the  use  of  the  ordinary  slide  rule  successfully  without  previous  experience 
and  without  knowledge  of  logarithms. 

Multiplication  and  Division  with  a  (Theoretical)  Complete  Loga- 
rithmic Scale.  Consider  a  complete  logarithmic  scale  (D,  Fig.  1),  assumed 
to  extend  indefinitely  in  both  directions,  only  the  main  section,  from  1  to 
10,  however,  being  usually  available.  Note  that  the  divisions  within  the 
several  sections  are  indentical,  except  that  the  numeral  attached  to  each  divi- 
sion of  any  one  section  is  ten  times  the  numeral  attached  to  the  corresponding 
division  in  the  preceding  section.  [The  distances  laid  off  from  1  are  propor- 
tional to  the  logarithms  of  the  corresponding  numbers,  the  distance  from  1  to 
10  being  taken  as  unity.]  Consider  also  a  duplicate  scale,  C,  numbered  from 
1  to  10,  and  arranged  to  slide  along  the  fixed  scale  D  as  in  the  figures.  By 
means  of  such  a  scale  D,  and  slide  C,  any  two  numbers  between  1  and  10 
(and  hence  any  two  numbers  whatever,  with  proper  attention  to  the  decimal 
point)  can  be  multiplied  or  divided,  as  in  the  following  examples. 


THE  SLIDE  RULE  95 

To  MULTIPLY  4  BY  6.  In  Fig.  1,  starting  with  point  1  of  the  fixed  scale, 
run  the  eye  along  from  1  to  4;  then  set  the  1  of  the  slide  opposite  this  point 
4,  and  run  the  eye  forward  along  the  slide  from  1  to  6 ;  the  point  thus  reached  on 
the  fixed  scale  is  24,  which  is  equal  to  4  X  6.  This  process  gives  the  distance 
from  1  to  4  plus  the  distance  from  1  to  6,  and  is,  in  fact,  a  mechanical  method 
of  adding  the  logarithms  of  these  numbers;  hence  the  result  is  the  product 
of  the  numbers.  Conversely, 


4       5    6    7  891 


FIG.  1. 

To  DIVIDE  4  BY  6.  In  Fig.  2,  starting  with  the  point  1  of  the  fixed  scale, 
run  the  eye  along  from  1  to  4;  then  set  the  6  of  the  slide  opposite  the  point  4, 
and  run  the  eye  backward  along  the  slide  from  6  to  1 ;  the  point  thus  reached  on 
the  fixed  scale  is  0.667,  which  is  equal  to  4  -r-  6.  This  process  gives  the  dis- 
tance from  1  to  4  minus  the  distance  from  1  to  6;  and  is,  in  fact,  a  mechanical 
method  of  subtracting  the  logarithms  of  these  numbers;  hence  the  result  is 
their  quotient. 


3        4 


5    6    7  8  9  l|0  _______  20  ____  jQ_    40    50  60  70  »0  «  l-O 


____________  ^i  4  +  6  •  0  667 

FIG.  2. 

Multiplication  and  Division,  Using  Only  a  Single  Section  of  the 
Scale.  If  only  the  main  section  of  scale  D  is  available  (as  is  usually  the  case 
in  practice),  the  result  of  multiplication  may  fall  beyond  the  scale,  as  it  does 
in.  Fig.  1.  In  such  cases  divide  the  first  factor  by  10  before  beginning  to  multiply; 
this  will  bring  the  result  within  the  scale,  without  affecting  the  sequence  of 
digits. 

For  example,  to  multiply  4  by  6.  Having  found  that  the  setting  shown  in  Fig.  1 
is  not  successful,  reset  the  slide  as  in  Fig.  3,  with  10  instead  of  1  opposite  4;  run  the  eye 
backward  along  the  slide  from  10  to  1,  thus  reaching  the  (unrecorded)  point  correspond- 
ing to  4  -j-  10;  then,  continuing  from  "this  point,  run  the  eye  forward  along  the  slide 
from  1  to  6,  as  before;  the  point  finally  reached  on  the  main  scale  is  2.4,  which  has  the 
Bame  sequence  of  digits  as  the  required  value  24.  After  a  little  practice,  this  preliminary 
step  of  dividing  by  10  will  be  performed  almost  intuitively.  Whether  or  not  this  step 
is  necessary  in  any  given  case,  can  be  determined  only  by  trial. 

The  general  rule  for  multiplication  may  be  stated  as  follows,  if  pre- 
ferred: To  find  the  product  of  two  factors,  find  one  factor  on  the  fixed  scale; 
opposite  this,  set  (tentatively)  point  1  of  the  slide;  on  the  slide  find  the  sec- 
ond factor,  and  opposite  this  read  the  product  on  the  main  scale,  if  possible. 
If  the  product  falls  beyond  the  scale,  begin  over  again,  using  point  10  of  the 
slide  instead  of  point  1. 

In  division  also,  the  result  may  fall  beyond  the  main  section  of  the  scale, 
as  it  does  in  Fig.  2.  In  such  cases,  it  suffices  merely  to  multiply  the  result 
by  10  in  order  to  bring  it  within  the  scale;  this  will  not  affect  the  sequence  of 
digits. 


96 


ARITHMETIC 


For  example,  to  divide  4  by  6,  set  the  slide  as  in  Fig.  4,  and  follow  out  mentally  the 
steps  indicated  by  the  arrows.  It  will  be  noticed  that  the  supplementary  step  of  multi- 
plying by  10  is  performed  by  simply  running  the  eye  along  the  slide  from  1  to  10  without 
resetting  the  slide;  for  this  reason,  division  on  the  slide  rule  is  slightly  easier  than 
multiplication. 


5    6    7  6  9  l|p 


i  3        4      56769  l|o 


-*j      4(*tO)x6'U 


FIG.  3. 


FIG.  4. 


The  Ordinary  Mannheim  Slide  Rule  has  four  scales,  A,  B,  C,  D,  as 
shown  in  Fig.  5.  Scales.  C  and  D  are  essentially  the  same  as  the  C  and  D 
scales  described  above,  and  the  principle  just  explained  shows  how  they  are 
used  in  multiplication  and  division.  The  fact  that  the  D  scale  covers  only  the 
main  section  from  1  to  10  (all  decimal  points  being  omitted)  is  practically  no 
restriction  on  the  scope  of  the  scale,  as  is  seen  in  the  preceding  examples. 
A  runner  is  provided,  so  that  intermediate  positions  reached  in  the  course 
of  an  extended  computation  may  be  indicated  temporarily  on  the  scale  without 
the  necessity  of  reading  off  their  numerical  values.  The  best  runners  are 
those  which  have  no  side  frame  to  obscure  the  numerals. 


f  A  1 


•W 


FIG.  5. 

In  problems  involving  successive  multiplications  and  divisions,   arrange 
the  work  so  that  multiplication  and  division  are  performed  alternately. 
X  &  X  c 


For  example,  to  calculate 


-,  divide  the  product  a  X  &  by  d;  multiply  this 


dX  e 

quotient  by  c;  and  divide  this  product  by  e.     Each  operation  will  require  only  one  shift- 
ing either  of  the  slide  (for  multiplication)  or  of  the  runner  (for  division). 

To  multiply  a  number  of  different  quantities  by  a  constant  multiplier,  x,  set 
the  point  1  of  slide  opposite  x,  and  read,  by  aid  of  the  runner,  the  prod- 
ucts of  x  by  all  the  quantities  which  do  not  fall  beyond  the  scale;  then  reset 
the  slide,  setting  10  instead  of  1  opposite  x,  and  read  the  products  of  x  by  all 
the  remaining  quantities. 

To  divide  a  number  of  different  quantities  by  a  constant  divisor,  y,  first 
find  (by  the  slide  rule)  the  quotient  1  -i-  y,  and  then  use  this  as  a  constant 
multiplier. 

Scales  A  and  B  are  exactly  like  scales  C  and  D,  except  that  they  cover  two 
sections  of  the  complete  logarithmic  scale,  the  graduations  being  only  half 
as  fine.  Either  pair  of  scales  may  be  used  for  multiplication  and  division; 
C  and  D  give  more  accurate  readings,  but  have  the  disadvantage  that  in  the 
case  of  multiplication  the  slide  must  often  be  shifted  to  the  other  end  in  order 
to  keep  the  result  on  the  scale — an  inconvenience  which  is  not  present  when 
the  less  accurate  scales  A  and  B  are  employed. 

By  the  use  of  both  pairs  of  scales,  problems  in  squares  and  square  roots 
may  be  readily  solved;  for  every  number  on  A,  except  for  the  decimal  point, 
is  the  square  of  the  number  directly  below  it  on  D  (use  the  runner). 


COMPUTING  MACHINES-,  FINANCIAL  ARITHMETIC  97 

A  scale  of  sines,  tangents,  and  logarithms  is  often  printed  on  the  back  of 
the  slide.  For  further  details  concerning  the  use  of  the  slide  rule  in  various 
problems,  see  the  instruction  books  furnished  with  each  instrument:  Wm.  Cox, 
"Manual  of  the  Mannheim  Slide  Rule;"  F.  A.  Halsey,  "Manual  of  the  Slide 
Rule;"  etc. 

Other  Types  of  Slide  Rules.  The  duplex  slide  rule  ($5  to  $18  according 
to  length)  shows  on  one  face  the  regular  A,  B,  C,  D  scales,  and  on  the  other  face  the 
scales  A,  B',  C',  D  (where  B'  and  C'  are  the  same  as  B  and  C,  only  numbered  in  the  re- 
verse order),  with  a  runner  encircling  the  whole  scale.  This  arrangement  makes 
possible  the  solution  of  more  complicated  problems  with  fewer  settings  of  the  slide,  but 
if  the  rule  is  to  be  used  only  for  simple  problems,  the  multiplicity  of  scales  is  rather  con- 
fusing. Less  complicated  is  the  polyphase  rule,  which  is  like  a  Mannheim  rule  with 
the  addition  of  a  single  inverted  scale,  C",  printed  in  the  middle  of  the  slide.  The  log 
log  duplex  slide  rule  (10  in.,  $8)  is  especially  adapted  for  handling  complex  problems 
involving  fractional  powers  or  roots,  hyperbolic  logarithms,  etc.  A  number  of  circular 
slide  rules  are  on  the  market,  the  best  of  which  are  operated  by  a  milled  thumbnut, 
like  the  stem  wind  of  a  watch.  The  advantage  of  the  circular  rule,  aside  from  its  com- 
pact size  (some  models  are  scarcely  larger  than  a  watch),  lies  in  the  fact  that  the  scale 
is  endless,  so  that  the  slide  never  has  to  be  reset  in  order  to  bring  the  result  within  the 
scale.  A  disadvantage  is  found  in  the  necessity  of  reading  the  figures  in  oblique  positions, 
or  else  continually  turning  the  instrument  as  a  whole  in  the  hand.  The  Fuller  and 
Thacher  rules  already  mentioned  are  invaluable  for  problems  requiring  greater  accuracy 
than  can  be  obtained  with  the  ordinary  rules.  There  are  also  many  special  slide  rules, 
adapted  to  various  special  types  of  computation,  such  as  calculating  discharge  of  water 
through  pipes,  horse  power  of  engines,  dimensions  of  lumber,  stadia  measurements,  etc. 
One  of  the  most  recent  devices  of  this  kind  is  the  Ross  meridiograph  (L.  Ross,  San 
Francisco,  Cal.),  which  is  a  circular  slide  rule  for  solving  certain  cases  of  spherical 
triangles.  The  Eichhorn  trigonometrical  slide  rule  solves  any  plane  triangle. 

COMPUTING  MACHINES 

For  certain  purposes  computing  machines  have  ceased  to  be  luxuries  and 
have  become  almost  necessities;  but  they  are  expensive,  and  should  be  selected 
with  reference  to  the  special  work  which  is  to  be  done.  The  machines  may 
be  classified  roughly  into  three  groups,  as  follows: 

Adding  Machines,  Non-listing.  Of  the  machines  of  this  kind,  the  most  convenient 
in  the  hands  of  a  careful  operator  is  the  well-known  Comptometer  (Felt  &  Tarrant  Co., 
Chicago,  111.;  $250  to  $350  according  to  size),  or  the  recent  Burroughs  non-listing 
adding  machine  (Detroit,  Mich.,  $175).  To  add  a  number,  simply  press  a  key  in 
the  proper  column;  the  result  appears  on  the  dials  in  front  of  the  keyboard.  Multi- 
plication as  well  as  addition  can  be  performed  on  this  machine  with  great  rapidity, 
and  division  also  after  a  little  practice.  Weight,  about  15  Ib.  Much  less  rapid,  but 
less  expensive  and  requiring  somewhat  less  skill  in  operation,  is  the  Barrett  adding 
machine  (Philadelphia,  Pa.)  with  multiplying  attachment.  Other  key-operated 
machines  are  the  Mechanical  Accountant  (Providence,  R.  I.),  and  the  Austin 
(Baltimore,  Md.).  The  American  adding  machine  (American  Can  Co.,  Chicago, 
111.;  $39,50)  is  operated  by  pulling  up  a  finger-lever  for  each  digit.  Small  machines, 
operated  by  the  use  of  a  stylus,  are  the  Rapid  computer  (Benton  Harbor,  Mich.,  $25); 
the  Gem  (Automatic  Adding  Machine  Co.,  New  York;  $10),  the  Arithstyle  (New 
York,  $36)  and  the  Triumph  (Brooklyn,  N.  Y.,  $35).  These  machines,  while  much 
less  rapid  than  the  key-operated  machines,  are  useful  in  simple  addition.  The  Under- 
wood typewriter  is  now  supplied  with  a  complete  electrically  driven  adding 
machine  attached,  and  the  Wahl  adding  attachment  is  supplied  on  the  Rem- 
ington and  other  typewriters.  Ray  Subtracto-Adder  (Richmond,  Va.f  $25). 

Adding  and  Listing  Machines.  The  machines  of  this  group  not  only  add,  but  also 
print  the  items,  totals  and  sub-totals.  The  Burroughs  (Detroit,  Mich.),  the  Wales  (Ad- 
der Machine  Co.,  Wilkes-Barre,  Pa.),  the  Comptograph  (Chicago,  111.)  and  the  White 
(New  Haven,  Conn.),  resemble  each  other  in  having  an  81-key  keyboard;  the  Dalton 
(Cincinnati,  Ohio)  and  the  Commercial  (White  Adding  Machine  Co.,  New  Haven, 


98  ARITHMETIC 

Conn.)  have  a  10-key  and  a  9-key  keyboard  respectively,  admitting  of  operation  by  the 
touch  method.  On  all  these  machines,  in  order  to  add  a  number,  first  depress  the  proper 
keys  and  then  pull  a  handle  (or,  in  the  case  of  electrically  driven  machines,  press  a 
button)  to  record  the  item.  Multiplication  cannot  be  performed  conveniently,  except 
on  the  Dalton.  Subtraction  can  be  performed  only  by  adding  the  complement,  except  on 
the  Commercial  and  on  one  type  of  the  Burroughs.  The  prices  range  from  $125  to 
$600,  according  to  size  and  style,  new  models  being  constantly  devised  for  special  com- 
mercial purposes.  A  new  and  more  portable  machine  of  the  81-key  type  is  the  Barrett 
adding  and  listing  machine  (Philadelphia,  Pa.,  $250).  A  cheaper  machine,  with  a  10- 
key  keyboard,  is  the  Standard  (St.  Louis,  Mo.).  The  new  American  adding  and 
listing  machine  (American  Can  Co.,  Chicago,  111.),  operated  by  pulling  up  a  finger-lever 
for  each  digit,  costs  only  $88.  The  Ellis  (Newark,  N.  J.)  is  an  elaborate  adding  and 
listing  machine  having  a  complete  typewriter  incorporated  with  it.  The  Elliott-Fisher 
bookkeeping  machine  (Harrisburg,  Pa.)  and  the  Moon-Hopkins  billing  ma- 
chine (St.  Louis,  Mo.)  are  intended  primarily  for  commercial  use;  the  latter  is  a  com- 
plicated electric  machine  ($750)  which  combines  many  of  the  features  of  an  adding  and 
listing  machine  with  those  of  a  calculating  machine. 

Calculating  Machines  (so-called).  Machines  of  this  third  group  are  intended 
primarily  for  multiplication  and  division;  the  types  which  have  a  keyboard  can  be 
used  effectively  for  addition  and  subtraction  also.  They  are  all  non-listing.  The 
earliest  commercially  successful  types  were  the  Thomas  and  the  Brunsviga.  In  both 
these  types  the  multiplicand  is  set  up  by  moving  pegs  in  slots,  or  (in  the  newest 
models)  by  depressing  keys,  and  the  multiplication  is  effected  by  turning  a  handle  for 
each  digit  of  the  multiplier — twice  for  a  digit  2,  three  times  for  a  digit  3,  etc. ;  the  result 
then  appears  on  the  dials.  In  the  Thomas  type  the  handle  always  turns  in  the  same  di- 
rection, the  change  from  multiplication  to  division  being  effected  by  a  shift  key.  In  the 
Brunsviga  type  the  handle  is  turned  forward  for  multiplication  and  backward  for  divi- 
sion. Among  the  best  examples  of  the  Thomas  type  now  on  the  American  market  are 
the  Tim,  with  a  single  row  of  dials,  the  Unitas,  with  a  double  row  of  dials  (both  sold 
by  Oscar  Miiller  Co.,  New  York  City;  also  with  keyboard  and  electric  drive),  and  the 
Reuter  (Philadelphia,  Pa.).  Prices,  $300  upward.  Another  machine  of  this  type, 
with  keyboard,  is  the  Record  (U.  S.  Adding  Machine  Co.,  New  York  City).  The 
Brunsviga  is  represented  by  Carl  H.  Reuter,  Philadelphia,  Pa. ;  various  models.  Of 
somewhat  similar  type  are  the  Triumphator  (New  York  City;  $250),  and  Colt's 
calculator  (Culmer  Engineering  Co.,  New  York  City).  A  new  machine,  on  the  same 
principle,  but  with  keyboard,  is  the  Monroe  (made  in  Orange,  N.  J.;  $250).  The 
Millionaire  (W.  A.  Morschhauser,  New  York  City;  $400),  is  from  the  mechanical  point 
of  view,  the  only  true  multiplying  machine  on  the  market  (except  the  Moon-Hopkins). 
After  the  multiplicand  is  set  up  on  the  pegs,  the  digits  of  the  multiplier  are  indicated 
successively  by  moving  a  pointer,  the  handle  being  turned  only  once  for  each  digit. 
Further,  the  movement  of  the  carriage  is  automatic.  The  newest  models  have  key- 
board and  electric  drive.  The  Ensign  electric  calculating  machine  (Boston,  Mass. ; 
$400)  is  a  new  machine  with  an  81-key  keyboard  on  which  it  adds  like  an  adding 
machine,  and  a  secondary  10-key  keyboard  by  means  of  which  it  multiplies  and 
divides  quite  as  rapidly  as  any  of  the  calculating  machines,  the  proper  key  being 
pressed  just  once  for  each  digit  of  the  multiplier.  The  National  calculator  (New 
York),  and  the  Lamb  calculator  (Calculator  Mfg.  Co.,  New  York)  are  less  ex- 
pensive machines  devised  for  figuring  payrolls  and  labor  costs.  A  still  simpler  device 
for  the  same  purpose  is  the  Calculacard  (New  York).  The  machine  called  the 
Calculagraph  (New  York)  is  a  time  clock  which  automatically  computes  labor  costs. 

For  graphical  methods  of  computation,  see  pp.  106,  119,  170,  173-185. 

FINANCIAL  ARITHMETIC 

For  the  facts  which  are  commonly  required  in  regard  to  compound  interest, 
sinking  funds,  etc.,  see  the  headings  of  the  tables  on  pp.  64-68. 


ELEMENTARY  GEOMETRY  AND  MENSURATION 


GEOMETRICAL  THEOREMS 

(For  geometrical  constructions,  see  p.  101) 

Right  Triangles,  a2  +  b2  =  c2.  (See  Fig.  1).  /A-f-^£=90°. 
2?2  =  mn.  a2  =  me.  62  =  nc.  See  also  p.  105  and  p.  132. 

Oblique  Triangles.  (See  also  pp.  105,  134.)  Sum  of  angles  =  180°.  An 
exterior  angle  =  sum  of  the  two  opposite  interior  angles.  (Fig.  1.) 

The  medians,  joining  each  vertex  with  the  middle  point  of  the  opposite  side, 
meet  in  the  center  of  gravity  G  (Fig.  2),  which  trisects  each  median. 

The  altitudes  meet  in  a  point  called  the  orthocenter,  0. 

The  perpendiculars  erected  at  the  midpoints  of  the  sides  meet  in  a  point 
C,  the  center  of  the  circumscribed  circle.  [In  any  triangle  G,  Ot  and  C  lie 
in  line,  and  G  is  two-thirds  of  the  way  from  O  to  C.] 


FIG.  1. 


FIG.  2. 


The  bisectors  of  the  angles  meet  in  the  center  of  the  inscribed  circle  (Fig.  3). 
The  largest  side  of  a  triangle  is  opposite  the  largest  angle;  it  is  less  than 
the  sum  of  the  other  two  sides,  and  greater  than  their  difference. 


FIG.  3. 


FIG.  4. 


Similar  Figures.  Any  two  similar  figures,  in  a  plane  or  in  space,  can  be 
placed  in  "perspective,"  that  is,  so  that  straight  lines  joining  corresponding 
points  of  the  two  figures  will  pass  through  a  common  point  (Fig.  4).  That  is, 
of  two  similar  figures,  one  is  merely  an  enlargement  of  the  other.  Assume 
that  each  length  in  one  figure  is  k  times  the  corresponding  length  in  the  other; 
then  each  area  in  the  first  figure  is  k2  times  the  corresponding  area  in  the  second, 
and  each  volume  in  the  first  figure  is  fc3  times  the  corresponding  volume 
in  the  second.  If  two  lines  are  cut  by  a  set  of  parallel  lines  (or  parallel  planes) , 
the  corresponding  segments  are  proportional. 

The  Circle.  (See  also  pp.  106,  137.)  An  angle  inscribed  in  a  semicircle 
is  a  right  angle  (Fig.  5).  An  angle  inscribed  in  a  circle,  or  an  angle  between 
a  chord  and  a  tangent,  is  measured  by  half  the  intercepted  arc  (Fig.  6).  An 
angle  formed  by  any  two  lines  which  meet  a  circle  is  measured  by  half  the 
sum  or  half  the  difference  of  the  intercepted  arcs,  according  as  the  point  of 
intersection  of  the  lines  lies  inside  (Fig.  7)  or  outside  the  circle  (Fig.  8). 

A  tangent  is  perpendicular  to  the  radius  drawn  to  the  point  of  contact. 

If  a  variable  line  through  A  (Figs.  9  and  10)  cuts  a  circle  in  P  and  Q,  then 


100 


ELEMENTARY  GEOMETRY  AND  MENSURATION 


AP  X  AQ   is   constant;  in  particular,  if  A  is   an  external  point,  AP  X  AQ 
-  AT2,  where  AT  is  the  tangent  from  A. 

T 


FIG.  5. 


Fio. 


FIG.  7. 


FIG.  8. 


FIG.  9.        FIG.  10. 


The  radical  axis  (Fig.  11)  of  two  circles  is  a  straight  line  such  that  the 
tangents  drawn  from  any  point  of  this  line  to  the  two  circles  are  of  equal 
length.  If  the  two  circles  intersect,  the  radical  axis  passes  through  their 
points  of  intersection.  In  any  case,  the  radical  axis  bisects  the  common 
tangents  of  the  two  circles.  The  three  radical  axes  of  a  set  of  three  circles 
meet  in  a  common  point.  (For  equations,  see  p.  137.) 


FIG. 


Dihedral  Angles.  The  dihedral  angle  between  two  planes  is  measured 
by  a  plane  angle  formed  by  two  lines,  one  in  each  plane,  perpendicular  to  the 
edge  (Fig.  12).  (For  solid  angles,  see  p.  110.) 

In  a  tetrahedron,  or  triangular  pyramid,  the  four  medians,  joining  each 
vertex  with  the  center  of  gravity  of  the  opposite  face,  meet  in  a  point,  the 
center  of  gravity  of  the  tetrahedron;  this  point  is  %  of  the  way  from  any 
vertex  to  the  center  of  gravity  of  the  opposite  face.  The  four  perpendiculars 
erected  at  the  circumcenters  of  the  four  faces  meet  in  a  point,  the  center  of 
the  circumscribed  sphere.  The  four  altitudes  meet  in  a  point  called  the 
orthocenter  of  the  tetrahedron.  The  planes  bisecting  the  six  dihedral 
angles  meet  in  a  point,  the  center  of  the  inscribed  sphere. 


FIG.  12. 


FIG. 


FIG.  14.      FIG.  15. 


FIG.  16. 


FIG.  17. 


Regular  Polyhedra  (see  also  p.  110):  Regular  tetrahedron  (Fig.  13), 
bounded  by  four  equilateral  triangles;  cube  (Fig.  14),  bounded  by  six  squares; 
octahedron  (Fig.  15),  bounded  by  eight  equilateral  triangles;  dodecahedron 
(Fig.  16),  bounded  by  twelve  regular  pentagons;  icosahedron  (Fig.  17), 
bounded  by  twenty  equilateral  triangles.  Figs.  13-17  show  how  these  solids 
can  be  made  by  cutting  the  surface  out  of  paper  and  folding  it  together. 

The  Sphere.  (See  also  p.  109.)  If  AB  is  a  diameter,  any  plane  perpen- 
dicular to  AB  cuts  the  sphere  in  a  circle,  of  which  A  and  B  are  called  the 
poles.  A  great  circle  on  the  sphere  is  formed  by  a  plane  passing  through 
the  center.  A  spherical  triangle  is  bounded  by  arcs  of  great  circles  (see  p. 


GEOMETRICAL  CONSTRUCTIONS  101 

134).  In  two  polar  triangles,  each  angle  in  one  is  tntf  fcuppfeme'nt'of  the 
corresponding  side  in  the  other.  In  two  symmetrical  triangles,  the  sides  and 
angles  of  one  are  equal  to  the  corresponding  sides  and  angles  of  the  other, 
but  arranged  in  the  reverse  order  (like  right-handed  and  left-handed  gloves). 

GEOMETRICAL  CONSTRUCTIONS 

To  Bisect  a  Line  AB  (Fig.  18).  (a)  From  A  and  B  as  centers,  and  with 
equal  radii,  describe  arcs  intersecting  in  P  and  Q,  and  draw  PQ,  which  will 
bisect  AB  in  M. 

(6)  Lay  off  AC  =  BD  =  approximately  half  of  AB,  and  then  bisect  CD. 

To  Draw  a  Parallel  to  a  Given  Line  1  Through  a  Given  Point  A  (Fig.  19). 
With  point  A  as  center  draw  an  arc  just  touching  the  line  Z;  with  any  point 
O  of  the  line  as  center,  draw  an  arc  BC  with  the  same  radius.  Then  a  line 
through  A  touching  this  arc  will  be  the  required  parallel.  Or,  use  a  straight 
edge  and  triangle.  Or,  use  a  sheet  of  celluloid  with  a  set  of  lines  parallel  to 
one  edge  and  about  y\  in.  apart  ruled  upon  it. 


>Q  7-X      (a.)  (b.) 

FIG.  18.  FIG.  19.  FIG.  20. 

To  Draw  a  Perpendicular  to  a  Given  Line  from  a  Given  Point  A 
Outside  the  Line  (Fig.  20).  (a)  With  A  as  center,  describe  an  arc  cutting 
the  line  in  R  and  S,  and  bisect  RS  in  M .  Then  M  is  the  foot  of  the  perpen- 
dicular. (&)  If  A  is  nearly  opposite  one  end  of  the  line,  take  any  point  B 
of  the  line  and  bisect  AB  in  O ;  then  with  O  as  center,  and  OA  or  OB  as  radius, 
draw  an  arc  cutting  the  line  in  M .  Or,  (c)  use  a  straight  edge  and  triangle. 


/?'  P    ^5  F*^ ~7Q  P       4         B 

FIG.  21.  FiGT22.  FIG.  23. 

To  Erect  a   Perpendicular   to  a  Given  Line   at  a  Given  Point  P. 

(a)  Lay  off  PR  =  PS  (Fig.  21),  and  with  R  and  S  as  centers  draw  arcs  inter- 
secting at  A.  Then  PA  is  the  required  perpendicular.  (6)  If  P  is  near  the 
end  of  the  line,  take  any  convenient  point  O  (Fig.  22)  above  the  line  as  center, 
and  with  radius  OP  draw  an  arc  cutting  the  line  in  Q.  Produce  QO  to  meet 
the  arc  in  A ;  then  PA  is  the  required  perpendicular,  (c)  Lay  off  PB  =  4 
units  of  any  scale  (Fig.  23) ;  from  P  and  B  as  centers  lay  off  PA  =  3  and 
BA  =  5;  then  APE  is  a  right  angle. 

To  Divide  a  Line  AB  into  n  Equal  Parts  (Fig.  24).  Through  A  draw 
a  line  AX  at  any  angle,  and  lay  off  n  equal  steps  along  this  line.  Connect 
the  last  of  these  divisions  with  B,  and  draw  parallels  through  the  other  divi- 


102 


ELEMENTARY  GEOMETRY  AND  MENSURATION 


cdori*.  These  parallels  wili  divide  the  given  line  into  n  equal  parts.  A  similar 
method  may  be  used  to  divide  a  line  into  parts  which  shall  be  proportional 
to  any  given  numbers. 


/ 

/ 

^u 

•  \    /-r 

,'  m 

\          ,,--'          Ix.,     I 

A            I 

3                           C       A                        A              fl 

FIG.  25.                          FIG.  26. 

FIG.  24. 

To  Construct  a  Mean  Proportional  (or  Geometric  Mean)  Between 
Two  Lengths,  m  and  n  (Fig.  25).  Lay  off  .AB  =  m  and  BC  =  n  and 
construct  a  semicircle  on  AC  as  diameter.  Let  the  perpendicular  erected  at 
B  meet  the  circumference  at  P.  Then  BP  =  -\/mn.  (See  p.  115.) 

To  Divide  a  Line  AB  in  Extreme  and  Mean  Ratio  (the  "golden  sec- 
tion"). At  one  end,  B,  of  the  given  line  (Fig.  26),  erect  a  perpendicular,  BO, 
equal  to  half  AB,  and  join  OA.  Along  OA  lay  off  OP  =  OB,  and  along  AB 
lay  off  AX  =  AP.  Then  X  is  the  required  point  of  division;  that  is,  AX2  = 
AB  XBX.  Numerically,  AX  =  ^(\/5 '-  1)(AB)  =  0.618(AB). 

To  Bisect  an  Angle  AOB  (Fig.  27).  Lay  off  OA  =  OB.  From  A  and  B 
as  centers,  with  any  convenient  radius,  draw  arcs  meeting  in  M ;  then  OM 
is  the  required  bisector. 

To  draw  the  bisector  of  an  angle  when  the  vertex  of  the  angle  is  not 
accessible  (Fig.  28).  Parallel  to  the  given  lines  a,  6,  and  equidistant  from 
them,  draw  two  lines  a',  &'  which  intersect;  then  bisect  the  angle  between  a' 
and  6'. 

-<*  PL. 

,-'Ct' 


FIG.  27. 


FIG.  28. 


To  Draw  a  Line  Through  a  Given  Point  A  and  in  the  Direction  of 
the  Point  of  Intersection  of  Two  Given  Lines,  when  this  point  of  inter- 
section is  inaccessible  (Fig.  29).  Draw  any  two  parallel  lines  PQ  and  P'Q' 
as  in  the  figure;  through  P'  draw  a  line  parallel  to  PA,  and  through  Q'  draw  a 
line  parallel  to  QA;  let  these  lines  intersect  in  A',  and  draw  the  line  AA'. 
This  line  A  A'  will  (if  produced)  pass  through  the  intersection  of  the  two 
given  lines. 

To  Construct,  Approximately,  the  Length  of  a  Circular  Arc  (Rankine). 
In  Fig.  30  draw  a  tangent  at  A.  Prolong  the  chord  BA  to  C,  making  AC  = 
H  AB.  With  C  as  center,  and  radius  CB, . 
draw  arc  cutting  the  tangent  in  D.  Then 
AD  =  arc  AB,  approximately  (error  about  4 
min.  in  an  arc  of  60  deg.).  Conversely,  to 
find  an  arc  AB  on  a  given  circle  to  equal  a 
given  length  AD,  take  E  one-fourth  of  the 
way  from  A  to  D,  and  with  E  as  center  and 
radius  ED  draw  an  arc  cutting  the  circum- 
ference in  B.  Then  arc  AB  =  AD,  approxi- 
mately. 


FIG.  30. 


GEOMETRICAL  CONSTRUCTIONS 


103 


To  Inscribe  a  Hexagon  in  a  Circle   (Fig.  31).     Step  around  the  cir- 
cumference with  a  chord  equal  to  the  radius.     Or,  use  a  60-deg.  triangle. 

To  Circumscribe  a  Hexagon 
About  a  Circle  (Fig.  32).  Draw 
a  chord  AB  equal  to  the  radius. 
Bisect  the  arc  AB  in  T.  Draw 
the  tangent  at  T  (parallel  to  AB), 
meeting  OA  and  OB  in  P  and  Q. 
Then  draw  a  circle  with  radius 
OP  or  OQ  and  inscribe  in  it  a  hex-  FIG.  31.  FIG.  32.  FIG.  33. 

agon,  one  side  being  PQ. 

To  Inscribe  an  Octagon  in  a  Square  (Fig.  33).  From  the  corners  as 
centers,  and  with  radius  equal  to  half  the  diagonal,  draw  four  arcs,  cutting 
the  sides  in  eight  points.  The  points  will  be 
the  vertices  of  the  octagon. 

To  Inscribe  an  Octagon  in  a  Circle.  Draw 
two  perpendicular  diameters,  and  bisect  each 
of  the  quadrant  arcs. 

To  Circumscribe  an  Octagon  About  a 
Circle.  Draw  a  square  about  the  circle,  and 
draw  the  tangents  to  the  circle  at  the  points 
where  the  circle  is  cut  by  the  diagonals  of  the 
square. 

To  Construct  a  Polygon  of  n  Sides,  One  A  8 

Side  AB  being  Given  (Fig.  34).     With  A  as  FIG.  34. 

center  and   AB  as  radius,  draw  a  semicircle, 

and  divide  it  into  n  parts,  of  which  n  —  2  parts  (counting  from  B)  are  to  be 
used.  Draw  rays  from  A  through  these  points  of  division,  and  complete  the 
construction  as  in  the  figure  (in  which  n  =  7).  Note 
that  the  center  of  the  polygon  must  lie  in  the  perpen- 
dicular bisector  of  each  side. 

To  Draw  a  Tangent  to  a  Cir- 
cle from  an  external  point  A  (Fig. 
35).  Bisect  AC  in  M;  with  M  as 
center  and  radius  MC,  draw  arc 
cutting  circle  in  P;  then  P  is  the  FIG.  35.  FIG.  36. 

required  point  of  tangency. 

To  Draw  a  Common  Tangent  to  Two  Given  Circles  (Fig.  36).  Let 
C  and  c  be  the  centers  and  R  and  r  the  radii  (R  >  r).  From  C  as  center,  draw 
two  concentric  circles  with  radii  R  +  r 
and  R  —  r;  draw  tangents  to  these 
circles  from  c;  then  draw  parallels  to 
these  lines  at  distance  r.  These  paral- 
lels will  be  the  required  common  tan- 
gents. 

To  Draw  a  Circle  Through  Three 
Given  Points  A,  B,  C,  or  to  find  the 
center  of  a  given  circular  arc  (Fig.  37).  FIG.  37.  FIG.  38. 

Draw  the  perpendicular   bisectors  of 
AB  and  BC;  these  will  meet  in  the  center,  0. 

To  Draw  a  Circular  Arc  Through  Three  Given  Points  When  the 
Center  is  not  Available  (Fig.  38).     With  A  and  B  as  centers,  and  chord 


104 


ELEMENTARY  GEOMETRY  AND  MENSURATION 


AB  as  radiua,  draw  arcs,  cut  by  BC  in  R  and  by  AC  in  S.  Divide  R A  into 
n  equal  parts,  1,  2,  3,  .  .  .  Divide  BS  into  the  same  number  of  equal  parts, 
and  continue  these  divisions  at  1',  2',  3',  ...  Connect  A  with  1',  2',  3',  .  . 
and  B  with  1,  2,  3,  .  .  . 
Then  the  points  of  intersec- 
tion of  corresponding  lines 
will  be  points  of  the  re- 
quired arc.  (Construction 
valid  only  when  CA  =  CB.) 

To    Draw    a   Circle 
Through    Two     Given 
Points,  A,  B,  and  Touch- 
ing a  Given  Line,  1  (Fig.  FlQ>  39  FlG.  40 
39).     Let  AB  meet  line  I  in 

C.  Draw  any  circle  through  A  and  B,  and  let  CT  be  tangent  to  this  circle 
from  C.  Along  Z,  lay  off  CP  and  CQ  equal  toCT.  Then  either  P  or  Q  is  the 
required  point  of  tangency.  (Two  solutions.)  Note  that  the  center  of  the 
required  circle  lies  in  the  perpendicular 
bisector  of  AB. 

To  Draw  a  Circle  Through  One  Given 
Point,  A,  and  Touching  Two  Given 
Lines,  1  and  m  (Fig.  40).  Draw  the 
bisector  of  the  angle  between  I  and  m,  and 
let  B  be  the  reflection  of  A  in  this  line. 
Then  draw  a  circle  through  A  and  B  and 
touching  I  (or  m},  as  in  preceding  con- 
struction. (Two  solutions.) 

To  Draw  a  Circle  Touching  Three 
Given  Lines  (Fig.  41).  Draw  the  bisec- 
tors of  the  three  angles;  these  will  meet  in 
the  center  O.  (Four  solutions.)  The 
perpendiculars  from  O  to  the  three  lines 
give  the  points  of  tangency. 

To  Draw  a  Circle  Through  Two  Given  Points  A,  B,  and  Touching 
a  Given  Circle  (Fig.  42).  Draw  any  circle  through  A  and  B,  cutting  the 
given  circle  in  C  and  D.  Let  AB  and  CD  meet  in  E,  and  let  ET  be  tangent 
from  E  to  the  circle  just 
drawn.  With  E  as  center, 
and  radius  ET,  draw  an 
arc  cutting  the  given  circle 
in  P  and  Q.  Either  P  or 
Q  is  the  required  point  of 
contact.  (Two  solutions.) 

To  Draw  a  Circle 
Through  One  Given 
Point,  A,  and  Touching 
Two  Given  Circles  (Fig. 
43).  Let  S  be  a  center  of 

similitude  for  the  two  given  circles,  that  is,  the  point  of  intersection  of  two 
external  (or  internal)  common  tangents.  Through  -S  draw  any  line  cutting 
one  circle  in  two  points,  the  nearer  of  which  shall  be  called  P,  and  the  other 
in  two  points,  the  more  remote  of  which  shall  be  called  Q.  Through  A,  P,  Q 


FIG.  41. 


FIG.  42. 


FIG.  43. 


LENGTHS  AND  AREAS  OF  PLANE  FIGURES 


105 


draw  a  circle  cutting  S A  in  B.  Then  draw  a  circle  through  A  and  B  and 
touching  one  of  the  given  circles  (see  preceding  construction).  This  circle 
will  touch  the  other  given  circle  also.  (Four  solutions.) 

To  Draw  an  Annulus  Which  Shall  Contain  a  Given  Number  of  Equal 
Contiguous  Circles  (Fig.  44).  (An  annulus  is  a 
ring-shaped  area  enclosed  between  two  concentric 
circles.)  Let  R  +  r  and  R  —  r  be  the  inner  and  outer 
radii  of  the  annulus,  r  being  the  radius  of  each  of  the 
n  circles.  Then  the  required  relation  between  these 
quantities  is  given  by  r  =  R  sin  (180°/n),  or  r  =  JTIG  44 

(R  +  r)[sin  (180%01/U  +  sin  (180%*)]. 

For  methods  of   constructing   ellipses   and   other    curves,    see   pp. 
139-156. 

LENGTHS  AND  AREAS  OF  PLANE  FIGURES 

Right  Triangle  (Fig.  45).     a2  +  62  =  c2. 

Area  =  tf  ab  =  Ma2  cot  A  =  H&2  tan  A  =  He2  sin  2 A. 

Equilateral  Triangle  (Fig.  46).     Area  =  Ma2\/3  =  0.43301a2. 


FIG.  45. 


a 
FIG.  47. 


Any  Triangle  (Fig.  47).     s  =  #  (a  +  b  +  c),   t  =  H(mx  +  m2 
r  =-\/(s  —  a)(s  —  6)(s  —  c)/s  =  radius  inscribed  circle, 
R  =  W  a/  sin  -A  =  },£&  /sin  B  =  ^c/sin  C  =  radius  circumscribed  circle; 
Area  =  H  base  X  altitude  =  \toh  =  tfab  sin  C  =  rs  =  abc/lR 


=  Vs(s  -o)(s  -&)(«  -  c)  =  ^  \*(*  -  mi)  (<  - 

=  r2  cot  ft  A  cot  H  B  cot  ^  C*    =  2#2  sin  A  sin  J5  sin  C 

=  ±  W  {  (zi2/2  —  £22/1)  +  (Z2Z/3  —  0:32/2)  +  (zsl/i  —  £12/3)  }  ,  where 

(zii  2/i).  (£2,  1/2),  (xs,  1/3)  are  co-ordinates  of  vertices.     See  also  p.  134. 


FIG.  48. 


FIG.  49. 


FIG.  50. 


Rectangle  (Fig.  48).  Area  =  ab  =  &D2  sin  u.  [u  =  angle  between 
diagonals  D,  D.] 

Rhombus  (Fig.  49).  Area  =  a2  sin  C  =  tfDiDz.  [C  =  angle  between 
two  adjacent  sides;  DI,  Dz  =  diagonals.] 

Parallelogram  (Fig.  50).  Area  =  bh  =  ab  sin  C  =  MDiD*  sin  u.  [u  = 
angle  between  diagonals  DI  andZ)2;Z>i2  +  Z>22  =  2(a2  +  62)]. 

Trapezoid  (Fig.  51).  Area  =  H(a  +  b)h  =  WDiDt  sin  u.  [Bases  a  and  b 
are  parallel;  u  =  angle  between  diagonals  DI  and  D*.] 


106 


ELEMENTARY  GEOMETRY  AND  MENSURATION 


sn  u  = 
c 


Quadrilateral  Inscribed  in  a  Circle  (Fig.  52).     Area 
V(s  -  a)(s  -  6)(s  -  c)(s  -  d)   =  tf(ac  -+  bd)sin  u\     s=  ft  (a 

Any  Quadrilateral  (Fig.  53).     Area  =  WDiDi   sin  u. 

NOTE,     a2  +  62  +  c2  +  dz  =  Z>i2  +  D22  +  4m2,  where  m  =  distance  between 
midpoints  of  D\  and  Dz. 

Polygons.     See  table,  p.  39. 


FIG.  52.  FIG.  53. 

Circle.     Area  =  -n-r2  =  ^O  =  y^Cd 
Here  r  =  radius,  d  =  diam.,  C  =  circumference  =  2irr  =  ird 

Annulus  (Fig.  54).     Area  =  ir(R2  -  r2)  =  7r(J>2  -  d2)/4 
R'  =  mean  radius  =  tf(R  +  r),  and  6  =  R  —  r. 

Sector  (Fig.  55).  Area  =  Mrs  =  7rr2(^/360°)  = 
y%rz  rad  A,  where  rad  A  =  radian  measure  of  angle 
A,  and  s  =  length  of  arc  =  r  rad  A  (table,  p.  44). 

Segment  (Fig.  56).     Area  =  J^r2  (rad  A  -  sin  A) 
=  yi[r(s  —  c)  +  ch],  where  rad  A  =  radian  measure  of 
angle  A  (table,  pp.  34-35,  44).     For  small  arcs, 
s  =  J.$(8c'  —  c),  where  c'  =  chord  of  half  the  are. 
(Huygens's  approximation.)    NOTE,  c  =  2\/h(d  —h) ; 
c'  =  \/dh  or  d  —  c'2/h,  where  d  =  diameter  of  circle ; 
h  =r  (1  —  cos  l/*tA),  s  —  2r  rad  %A. 

Ribbon  bounded  by  two  parallel  curves  (Fig.  57). 
If  a  straight  line  AB  moves  so  that  it  is  always  per- 
pendicular to  the  path  traced  by  its  middle  point  G, 


FIG.  55. 

=  0.785398d2  (table,  p.  30). 
(table,  p.  28). 
=  2-n-R'b,  where 


FIG.  56. 


FIG.  57. 


then  the  area  of  the  ribbon  or  strip  thus  generated  is  equal  to  the  length  of 
AB  times  the  length  of  the  path  traced  by  G.  (It  is  assumed  that  the  radius 
of  curvature  of  G's  path  is  never  less  than  tf  AB,  so  that  successive  positions 
of  the  generating  line  will  not  intersect.) 

Simpson's  Rule  (Fig.  58).  Divide  the 
given  area  into  n  panels  (where  n  is  some 
even  number)  by  means  of  n  +  1  parallel 
lines,  called  ordinates,  drawn  at  constant  dis- 
tance h  apart;  and  denote  the  lengths  of  these 
ordinates  by  3/0,  2/i,  2/2,  .  .  ,  2/n.  (Note  that 
2/o  or  yn  may  be  zero.)  Then 
Area  =  ^h[(y0  +  yn)  +  4(yi  +2/3+2/5.  .  •) 
+  2(2/2  +  2/4  +  2/6-  •  . )  ].  approx.  The  greater 


FIG.  58. 


the  number  of  divisions,  the  more  accurate  the  result.      Note:     Taking  y 
=  f(x) ,  where  x  varies  from  x  =  a  to  x  =  6,  and  h  =  (b  —  a)  /n,  then   the 

error  = ~  ^&  ~^  f""(X),  where  f""(X)  is  the  value  of  the  fourth  de- 

loO  71 

rivative  of  f(x)  for  some  (unknown)  value,  x  =  X,  between  a  and  6. 


SURFACES  AND   VOLUMES  OF  SOLIDS 


107 


Ellipse  (Fig.  59;  see  also  p.  140).     Area  of  ellipse  =  trab.     Area  of  shaded 
segment  =  xy  +  ab  sin"1  (x/a).     Length  of  perimeter  of  ellipse  =  ir(a  +  b)K, 
where  K  =  [1  +  Y*mz  +  H*m4  +  ^sem6  +  ...],     m  =  (a  -  6)/(a  +  6). 
Form  =0.1       0.2       0.3       0.4       0.5       0.6       0.7       0.8       0.9       1.0 
K=  1.002  1.010  1.023  1.040  1.064  1.092  1.127  1.168  1.216  1.273 


(«_.__  a  _..J 
FIG.  60. 
Hyperbola   (Fig.  60;  see  also  p.   144).     In  any  hyperbola,  shaded  area 

A  —  ab   loge  ( — (-")•     In    an    equilateral    hyperbola    (a  =  &),    area   A  = 

\a       b/ 

o8  sirih-l(y/d)  =  o2  cosh-l(x/d).     For  tables  of  hyperbolic  functions,  see  p.  60. 
Here  x  and  y  are  co-ordinates  of  point  P. 

Parabola  (Fig.  61;  see  also  p.  138).  Shaded  area  A  =  ftch.  In  Fig.  62, 
length  of  arc  OP  =  s  =  tyPT  +  typ  loge  cot  ^u.  Here  c  =  any  chord;  p  = 
semi-latus  rectum;  PT  =  tangent  at  P.  Note:  OT  =  OM  =  x. 


FIG.  61.  FIG.  62. 

Other  Curves.     For  lengths  and  areas,  see  pp.  147-156. 

SURFACES  AND  VOLUMES  OF  SOLIDS 

Regular  Prism  (Fig.  63).  Volume  =  ^nrah  =  Bh.  Lateral  area  = 
nah  =  Ph.  Here  n  =  number  of  sides;  B  =  area  of  base;  P  =  perimeter  of  base. 

Right  Circular  Cylinder  (Fig.  64).  Volume  =  irr2h  =  Bh.  Lateral 
area  =  lirrh  =  Ph.  Here  B  =  area  of  base;  P  =  perimeter  of  base. 


FIG.  63. 


FIG.  64. 


FIG.   65. 


FIG.  66. 


Truncated  Right  Circular  Cylinder  (Fig.  65).  Volume  =  -n-r2h  =  Bh. 
Lateral  area  =  2wrh  =  Ph.  Here  h  =  mean  height  =  tf(hi  +  ta) ;  B  =  area 
of  base;  P  =  perimeter  of  base. 


108 


ELEMENTARY  GEOMETRY  AND  MENSURATION 


Any  Prism  or  Cylinder  (Fig.  66).  Volume  =  Bh  =  NL  Lateral  area 
=  Ql.  Here  I  =  length  of  an  element  or  lateral  edge;  B  =  area  of  base;  N  = 
area  of  normal  section;  Q  =  perimeter  of  normal  section. 

Any  Truncated  Prism  or  Cylinder  (Fig.  67).  Volume  =  Nl.  Lateral 
area  =  Qk.  Here  I  =  distance  between  centers  of  gravity  of  areas  of  the  two 
bases;  k  =  distance  between  centers  of  gravity  of  perimeters  of  the  two  bases; 
N  =  area  of  normal  section;  Q  =  perimeter  of  normal  section.  For  a  trun- 
cated triangular  prism  with  lateral  edges a,b,c,  I  =  k  =  \i(a  +6  +  c).  Note: 
I  and  k  will  always  be  parallel  to  the  elements. 


FIG.  67. 


FIG. 


FIG.  69. 


FIG.  70. 


Special  Ungula  of  a  right  circular  cylinder.     (Fig.  68.).     Volume  = 
Lateral  area  =  2rH.     r  =  radius.     (Upper  surface  is  a  semi-ellipse.) 

Any  Ungula  of  a  right  circular  cylinder.  (Figs.  69  and  70.)  Volume  = 
H(Ha3  ±  c£)/(r  ±  c)  =  H[a(r*  -  ^a2)  ±  r2crad  u]/(r  ±  c).  Lateral  area  = 
H(2ra  ±  cs)/(r  ±  c)  =  2rH(a  ±  c  rad  u)/(r  ±  c).  If  base  is  greater  (less) 
than  a  semicircle,  use  +  (— )  sign,  r  =  radius  of  base;  B  =  area  of  base; 
s=  arc  of  base;  u=  half  the  angle  subtended  by  arc  s  at  center;  rad  u  = 
radian  measure  of  angle  u  (see  table,  p.  44). 

Hollow  Cylinder  (right 
and  circular).  Volume  = 
7rh(R*  -r2)  =Trhb(D-  b) 
=  Trhb(d  +  6)  =  MD'  = 
irhb  (R  +  r).  Here  h  = 
altitude;  r,R(d,D)  =  inner 
and  outer  radii  (diameters) ; 
b  =  thickness  =  R—  r; 
D'  =  mean  diam.  =ty(d  + 
D)  =D  -b  =d+6. 

Regular  Pyramid  (Fig. 
71).  Volume  =  1$  altitude 
X  area  of  base  —  ^hran. 
base  =  ^san.  Here  r  =  radius  of  inscribed  circle;  a  =  side  (of  regular 
polygon) ;  n  —  number  of  sides;  s  =  \/r*  +  h2.  Vertex  of  pyramid  directly 
above  center  of  base. 

Right  Circular  Cone.  Volume  =  H-irrzh.  Lateral  area  =  TTTS.  Here 
r  =*  radius  of  base;  h  =  altitude;  s  =  slant  height  =  \/r2  +  h2. 

Frustum  of  Regular  Pyramid  (Fig.  72). 

Volume  =  Uhran[l  +  (a'/o)  +  (a' /«)*]• 

Lateral  area  =  slant  height  X  half  sum  of  perimeters  of  bases  =  slant 
height  X  perimeter  of  mid-section  =  J/isn(r  +  r').  Here  r,r'  =  radii 


FIG.  71. 
Lateral  area 


FIG.  72.  FIG.  73. 

slant  height  X  perimeter  of 


SURFACES  AND  VOLUMES  OF  SOLIDS 


109 


of  inscribed  circles;  s  =  V  (r  —  r')2  +  ft2;   a,ar    =  sides  of  lower  and  upper 
bases;  n  =  number  of  sides. 

Frustum      of      Right      Circular      Cone      (Fig.      73).        Volume  = 

Lateral  area  =  irs(r  +  r')  ;>  =  V(r  -  r')2  +  ft2. 

Any  Pyramid  or  Cone.  Volume  =  ftBh.  B  =  area  of  base;  h  =  perpen- 
dicular distance  from  vertex  to  plane  in  which  base  lies. 

Any  Pyramidal  or  Conical  Frustum  (Fig.  74).  Volume  = 
HftCB  +  VBB~'  +  B")  =  XhB[l  +  (P'/P)  +  (P'/P)2]-  Here  B,  B'  =  areas  of 
lower  and  upper  bases;  P, P'  =  perimeters  of  lower  and  upper  bases. 


FIG.  74. 


FIG.  75. 


FIQ.  76. 


Obelisk  (Frustum  of  a  rectangular  pyramid.     Fig.  75). 

Volume  =  ^ft[(2a  +  ai)6  +  (2ai  +  a)  61]  =  Kh[ab  +  (a  +  ai)  (6  +  61)  + 

Wedge  (Rectangular  base;  ai  parallel  to  a,a  and  at  distance  ft  above  base. 
Fig.  76).  Volume  =  }6ft6(2a  +  ai). 

Sphere.  Volume  =  V  =  ^Trr3  =.  4.188790r3  =  H^3  =  0.523599d8  (table, 
p.  36)  =  %  volume  of  circumscribed  cylinder.  Area  =  A  —  4rnrz  =  four  great 
circles  (table,  p.  30)  =  ird2  =  3.14159d2  =  lateral  area  of  circumscribed  cylinder. 
Here  r  =  radius;  d  =  2r  =  diameter  =  ^/67/7r  =  1.24070  i/7 

0.56419\/Z- 

Hollow  Sphere,  or  spherical  shell.  Volume  = 
Hir(R3  —  r3)  =  H7r(D3  —  d8)  =  4irRiH  +  lint*.  Here 
-R,r  =  outer  and  inner  radii;  D,d  =  outer  and  inner 
diameters;  t  =  thickness  =  R  —  r;  RI  =  mean  radius  = 

Spherical  Segment  of  One  Base.  Zone  (spher- 
ical "cap"  of  Fig.  78).  Volume  =  H7rft(3o2  +  ft2)  = 
H7rft2(3r  —  ft)  (table,  p.  38).  Lateral  area  (of  zone)  = 
27rrft=  7r(a.2 +  ft2).  Note:  a2  =  ft(2r  -  ft),  where  r 
=  radius  of  sphere. 

Any  Spherical  Segment.  Zone  (Fig.  77).  Vol- 
ume =  ^?rft(3a2  +  3ai2  +  ft2).  Lateral  area  (zone) 
=  27rrft.  Here  r  =  radius  of  sphere.  If  the  inscribed 
frustum  of  a  cone  be  removed  from  the  spherical  seg- 
ment, the  volume  remaining  is  Mirhc2,  where  c  =  slant 
height  of  frustum  =  \/ft2  +  (p>  ~  °i)2- 

Spherical     Sector     (Fig.    78).     Volume  =  W  X  area 
Total    area  =  area    of    cap  +  area    of    cone 
ft(2r  -  ft). 


of    cap 
2irrh  +  vra.     Note:  o2 


110 


ELEMENTARY  GEOMETRY  AND  MENSURATION 


(Fig. 
area 


Spherical  Wedge  bounded  by  two  plane  semicircles  and  a  lune. 
79.)  Volume  of  wedge  -r-  volume  of  sphere  =  w/3600.  Area  of  lune 
of  sphere  =  w/360°.  u  =  dihedral  angle  of  the  wedge. 

Spherical  Triangle  bounded  by  arcs  of  three  great  circles.  (Fig.  80.) 
Area  of  triangle  =  Trr2E/l80°  =  area  of  octant  X  #/90°.  E  =  spherical 
excess  =  180°  —  (A  +  B  +  C),  where  A,  B,  and  C  are  angles  of  the  triangle. 
See  also  p.  134. 

Solid  Angles.  Any  portion  of  a  spherical  surface  subtends  what  is 
called  a  solid  angle  at  the  center  of  the  sphere.  If  the  area  of  the  given 
portion  of  spherical  surface  is  equal  to  the  square  of 
the  radius,  the  subtended  solid  angle  is  called  a 
steradian,  and  this  is  commonly  taken  as  the  unit. 
The  entire  solid  angle  about  the  center  is  called  a 
steregon,  so  that  4r  steradians  =  1  steregon.  A 
so-called  "solid  right  angle"  is  the  solid  angle  sub- 
tended by  a  quadrantal  (or  trirectangular)  spherical 
triangle,  and  a  "spherical  degree"  (now  little  used) 
is  a  solid  angle  equal  to  ^o  of  a  solid  right  angle. 
Hence  720  spherical  degrees  =  1  steregon,  or  TT  stera- 
dians =  180  spherical  degrees.  If  u  =  the  angle 

which  an  element  of  a  cone  makes  with  its  axis,  then  the  solid  angle  'of  the 
cone  contains  2?r(l  —  cos  u)  steradians. 

Regular  Polyhedra.     A  =  area  of  surface;  V  =  volume;  a  =  edge. 

Name  of  solid  (see  p.  100)  Bounded  by  A/a2  V/a* 

Tetrahedron  ...............................  4  triangles  1.7321  0.1179 

Cube  ......................................  6  squares  6.0000  1.0000 

Octahedron  ................................  8  triangles  3.4641  0.4714 

Dodecahedron  ............................  .  .  12  pentagons  20.  6457  7.  6631 

Icosahedron  ..............................  20  triangles  8.6603  2.1817 

Ellipsoid  (Fig.  81).     Volume  =  tynrdbc,  where  a,  b,  c  =  semi-axes. 

Spheroid  (or  ellipsoid  of  revolution).  The  volume  of  any  segment  made 
by  two  planes  perpendicular  to  the  axis  of  revolution  may  be  found  ac- 
curately by  the  prismoidal  formula  (p.  111). 


FIG.  79.       FIG.  80. 


FIG.  81. 


FIG.  82. 


FIG.  83. 


FIG.  84. 


Paraboloid  of  Revolution  (Fig.  82).  Volume  =  y^rr^h  =  ^  volume  of 
circumscribed  cylinder. 

Segment  of  Paraboloid  of  Revolution  (Bases  perpendicular  to  axis, 
Fig.  83).  Volume  of  segment  =  ynr(R*  +  rz}h. 

Barrels  or  Casks  (Fig.  84),  Volume  =  ],i2irh(2D2  +  d2)  approx.  for  cir- 
cular staves.  Volume  =  Wvrh(2D2  +  Dd  +  ^d2)  exactly  for  parabolic  staves. 


SURFACES  AND   VOLUMES  OF  SOLIDS 


111 


FIG.  85. 


For  a  standing  cask,  partially  full,  compute  contents 
by  the  prismoidal  formula,  p.  111.  Roughly,  the  num- 
ber of  gallons,  G,  in  a  cask  is  given  by  G  =  0.0034?i2/i, 
where  n  =  number  of  inches  in  the  mean  diameter, 
or  H(-D  +  d),  and  h  =  number  of  inches  in  the  height. 

Torus,    or    Anchor    Ring    (Fig.    85).     Volume  = 
2:r2cr2.     Area  =  4vr2cr  (Proof  by  theorems  of  Pappus). 

Theorems  of  Pappus.  1.  Assume  that  a  plane  figure,  area  A,  revolves 
about  an  axis  in  its  plane  but  not  cutting  it;  and  let  s  =  length  of  circular 
arc  traced  by  its  center  of  gravity.  Then  volume  of  the  solid  generated  by 
A  is  V  =  As.  For  a  complete  revolution,  V  —  2irrA,  where  r  =  distance 
from  axis  to  center  of  gravity  of  A. 

2.  Assume  that  a  plane  curve,  length  I,  revolves  about  an  axis  in  its  plane 
but  not  cutting  it;  and  lets  =  length  of  circular  arc  traced  by  its  center 
of  gravity.  Then  area  of  the  surface  generated  by  I  is  S  =  Is.  For  a 
complete  revolution,  S  =  2-irrl,  where  r  =  distance  from  axis  to  center  of 
gravity  of  I. 

NOTE.  If  Vi  or  Si  about  any  axis  is  known,  then  Vz  or  £2  about  any 
parallel  axis  can  be  readily  computed  when  the  distance  between  the  axes  is 
known. 

Generalized  Theorems  of  Pappus.  Consider  any  curved  path  of 
length  s.  If  (1)  a  plane  figure,  area  A  [or  (2)  a  plane 
curve,  length  I]  moves  so  that  its  center  of  gravity 
slides  along  this  curved  path  (Fig.  86),  while  the 
plane  of  A  [or  I]  remains  always  perpendicular  to  the 
path,  then  (1)  the  volume  generated  by  A  is  V  =  As 
[and  (2)  the  area  generated  by  I  is  S  —  Is].  The 
path  is  assumed  to  curve  so  gradually  that  successive  positions  of  A  [or  I] 
will  not  intersect. 

The  Prismoidal  Formula  (Fig.  87).  Volume  =Uh(A  +B  +.4M), 
where  h  =  altitude,  A  and  B  =  areas  of  bases  and  M  =  area  of  a  plane  section 
midway  between  the 
bases.  This  formula  is 
exactly  true  for  any 
solid  lying  between  two 
parallel  planes  and  such 
that  the  area  of  a  sec- 
tion at  distance  x  from  Fm>  g7  FlG>  gg 
one  of  these  planes  is 

expressible  as  a  polynomial  of  not  higher  than  the  third  degree  in  x. 
approximately  true  for  many  other  solids. 

Simpson's  Rule  may  be  applied  to  finding  volumes,  i'f  the  ordinates 
2/i,  2/2,  be  interpreted  as  the  areas  of  plane  sections,  at  constant  distance 
h  apart  (p.  106). 

Cavalieri's  Theorem.  Assume  two  solids  to  have  their  bases  in  the 
same  plane.  If  the  plane  section  of  one  solid  at  every  distance  x  above  the 
base  is  equal  in  area  to  the  plane  section  of  the  other  solid  at  the  same  dis- 
tance x  above  the  base,  then  the  volumes  of  the  two  solids  will  be  equal. 
See  Fig.  88. 


FIG.  86. 


It  is 


ALGEBRA 

FORMAL  ALGEBRA 

Notation.  The  main  points  of  separation  in  a  simple  algebraic  expres- 
sion are  the  +  and  —  signs.  Thus,,  a  +  b  X  c  —  d  +  x  -)-  y  is  to  be  inter- 
preted as  a  +  (b  X  c)  —  (d  -5-  3)  +  y.  In  other  words,  the  range  of  opera- 
tion of  the  symbols  X  and  -i-  extends  only  so  far  as  the  next^+  or  —  sign. 
As  between  the  signs  X  and  -f-  themselves,  a  -f-  b  X  c  means,  properly  speak- 
ing, a  -r-  (b  X  c);  that  is,  the  -5-  sign  is  the  stronger  separative;  but  this  rule 
is  not  always  strictly  followed,  and  in  order  to  avoid  ambiguity  it  is  better 
to  use  the  parentheses. 

The  range  of  influence  of  exponents  and  radical  signs  extends  only  over 
the  next,  adjacent  quantity.  Thus,  2ax3  means  2a(x3),  and  \^2ax  means 
(\/2)  (ax).  Instead  of  \/2ax,  it  is  safer,  however,  to  write  \/2'ax,  or,  bet- 

ter, ox's/2- 

Any  expression  within  parentheses  is  to  be  treated  as  a  single  quantity. 
A  horizontal  bar  serves  the  same  purpose  as  parentheses. 

The  notation  a-b,  or  simply  ab,  means  a  X  b;  and  a:  b,  or  a/b,  means  a  -f-  b. 

The  symbol  |a|  means  the  "absolute  value  of  a,"  regardless  of  sign;  thus, 
|-2|  =  |  +  2|  =2. 

The  symbol  nl  (where  n  is  a  whole  number)  is  read:  "n  factorial,"  and 
means  the  product  of  the  natural  numbers  from  1  to  n,  inclusive.  Thus 
1!  =  1;  2!  =  1  X  2;  3!  =  1  X  2  X  3;  4!  =  1  X  2  X  3  X  4  /etc. 

The  symbol  ^  or   =f  means,  "not  equal  to";  ±  means  "plus  or  minus." 

The  symbol  «  is  som'etimes  used  for  "approximately  equal  to." 

Addition      and     Subtraction,     a  +  b  =  b  +  a. 

(a  +  b)  +  c  =  a  +  (b  +  c).     a  -  (  -  b)  =  a  +  b.     a  -  a  =  0. 

a  +  (x  —  y  +  z)  =  a+x—  y+z.  a  —  (x^  —  y  +  z)  =  a  —  x  +  y—  z. 
A  minus  sign  preceding  a  parenthesis  operates'  to  reverse  the  sign  of  every 
term  within,  when  the  parentheses  are  removed. 

Multiplication  and  Simple  Factoring,  ab  =  ba.  (ab)c  =  a(bc). 
a(b  -f-  c)  =  ab  +  ac.  a(b  —  c)  =  ab  —  ac.  Also,  a  X  (  —  b)  =  —  ab,  and 
(  —  o)  X  (  —  b)  =  ab;  "unlike  signs  give  minus;  like  signs  give  plus." 

(a  +  6)  (a  -  6)  =  a2  -  b2. 

(a  +  b)2  ='a2  +  2ab  +  b2,      (a  -  6)2  =a2  -  2ab  +  62. 

(a  +  &)3  =  o3  +  3o2&  +  3a&2  +  63,  (a  -  6)3  =  a3  -3a2&+  3a&2  -  63;  etc. 
(See  table  of  binomial  coefficients,  p.  39;  also  p.  114.) 

02  _  &2  =  (o  -  &)(o  +  6),      a3  -  63  =  (o  -  6)  (a2  +  ab  +  b2). 

an  -  bn  =  (a  -  6)  (a71-1  +  an-26  +  an~362  +  .    .    .  +  abn~2  +  6"-1). 

an  +  bn  is  factorable  by  a  +  6  only  when  n   is  odd  ;  thus, 

aa  4.  &3  =  (O._j_  5)(a2  _  ab  +  b2), 

ae  +  &5  =  (0  _|_  6)  (o4  -  a3b  +  a2b2  -  ab3  +  b4)  ;     etc. 

The  following  transformation  is  sometimes  useful  : 


f  /         b  \  2       (Vb^^4a~c\  21 
ax*+bx+c=a[(x+-)    -(         ^         )   J. 

ma  +  mb  -\-rnc       a+b+c 
Fractions.     If    w    is    not    zero,    -  -  =  —  —  ;   that  is, 

mx  +  my  x  -\-  y 

both  numerator  and  denominator  of  a  fraction  may  be  multiplied  or  divided 

112 


FORMAL  ALGEBRA  113 

by  any  quantity  different  from   zero,  without   altering  the  value  of   the 
fraction. 

To  add  two  fractions,  reduce  each  to  a  common  denominator,  and  add  the 

a       x       ay       bx      ay  +  bx 
numerators:       r  +  ~  =  ;  --  H  r~  =  -  ;  -  • 

b       y      by      by  by 

a      x      ax      a  ax      ax 

To  multiply  two  fractions:  T  X  -  =  r~;    T  X  z  =  -r  X-  =  ~r. 
b      y      by      b  bib 

To  divide  one  fraction  by  another,  invert  the  divisor  and  multiply: 
o  ^5  _  ja  V2/  _ay     _a  _._       _  a       1  _  ^ 

t  •  ~~         T         -^  ~"~      1  >  1  •         X       —  /\  —  —  • 

b       y        b    .  x       bx       b  b       x       bx 

Ratio  and  Proportion.  The  notation  a:b:  :c:d,  which  is  now  passing 
out  of  use,  is  read:  "a  is  to  &  as  c  is  to  d,"  and  means  simply  (a/6)  =  (c/d), 
or  ad  =  be.  a  and  d  are  called  the  "extremes,"  6  and  c  the  "means," 
and  d  the  "fourth  proportional"  to  a,  b,  and  c.  The  "mean  proportional" 
between  two  numbers  is  the  square  root  of  their  product;  also  called  the 
"geometric  mean"  of  the  numbers  (p.  115).  If  a/6  ='  c/d,  then  (a  -{-&)/&  = 
(c  +  d)/d,  and  (a  -  &)/&  =  (c  —  d)/d;  whence  also,  (a  -\-  6)  /(a  -  6)  = 
(c+d)/(c-d).  I*a/x=b/y=c/z  =  .  .  .  =  r,  then 


Variation.  The  notation  x  cc  y  is  read:  "a;  varies  directly  as  y,"  or  "x 
is  directly  proportional  to  y,"  and  means  a;  =  ky,  where  k  is  some  constant. 
To  determine  the  constant  k,  it  is  sufficient  to  know  any  pair  of  values,  as 
x\  and  2/1,  which  belong  together;  then  o?i  =  kyi,  and  hence  x/xi  =  y/yi,  or 
£  =  (r<i/yi)y.  The  expression  "x  varies  inversely  as  y"  or  "x  is  inversely 
proportional  to  y,"  means  that  x  is  proportional  to  1/y,  or  x  =  A/y. 

Exponents.  om+n  =  awan.  aw~n  =am/an.  a°  =  1  (if  a  5^  0).  o~m  =  l/am. 
(a™)*  •=  a""1,  a1/"  =  AVa-  Thus:  aH  =_  A/a",  and  aM  =  ^ou  aw/n  - 
Va^.  Thus:a%  =  Va2  and  a%  =  Va*.  (V^)n  =  a.  (ab)n  =  anbn. 
(a/b)n  =  an/bn.  (—  a)n  =  an  if  n  is  even.  (—  a)n  =  —  an  if  n  is  odd. 
If  n  is  positive  and  increases  indefinitely,  an  becomes  infinite  if  a  >  1,  and 
approaches  0  if  a  <  1  (a  being  always  positive).  Graphs,  p.  174;  series,  p.  160. 

Radicals.     Except  in  the  simple  cases  of  square  root  and  cube  root,  radical 
signs    should   always   be   replaced   by   fractional    exponents:    "v/a  =  a   n. 
(Va)n  =  (a1/n)«  =  a.     If  n  is  odd,    \/^a  =  -  -\/a't   but  if   n  is   even, 
V  —  a  is  imaginary.     Every  positive  number  a  has  two  square  roots,  one 
positive  and  the  other  negative;  but  the  notation  \/a  always  means  the  positive 
root;  thus,  v9  =  3;_—  V9  =  —  3.     If  the  denominator  of  a  fraction  is  of 
the  form  va  +  v  b,  it  is  possible  to  "rationalize  the  denominator"  by 
multiplying  both  numerator  and  denominator  by  \/a  +  \/&.     Thus: 
A/a  +  A/6  =  (Va  +.Vb)(Va  +  A/6)  _  a  +  b  +  2A/a& 
A/a  -  A/6       (A/^  -  A/6)(Va  +  V&)  ~  0-6 

Logarithms.  (For  the  use  of  logarithms  in  numerical  computation, 
see  p.  91.)  The  logarithm  of  a  (positive)  number  N  is  the  exponent  of  that 
power  to  which  the  base  (10  or  e)  must  be  raised  to  produce  N.  Thus,  x 
=  logio  N  means  that  10*  =  N,  and  x  =  Iog8  N  means  that  ex  =  N.  Loga- 
rithms to  base  10  are  called  common,  denary,  or  Briggsian  logarithms. 
For  table  of  4-pIace  common  logarithms  see  pp.  40-43. 


114  ALGEBRA 

Logarithms  to  base  e  are  called  hyperbolic,  natural,  or  Napierian  logar- 
ithms. Here  e  =  1  +  1  +  1/2!  +  1  /3!  +  1/4!  +  .  .  .  =  2.718281828459.  .  . 
For  table  of  4-place  hyperbolic  logarithms  see  pp.  58,  59. 

If  the  subscript  10  or  e  is  omitted,  the  base  must  be  inferred  from  the 
context,  the  base  10  being  used  in  numerical  computation,  and  the  base  e 
in  theoretical  work.  In  either  system, 

log  (o&)    =  log  a  +  log  b  log  (an)        =  n  log  a  log  0  =   —  °° 

log  (0/6)   =  log  a  —  log  b  log  (V/a~)   =  (1/w)  log  a          log    1=0 

log  (1/n)   =   —  log  n  log   (base)   =  1  log  co  =  oo 

The  two  systems  are  related  as  follows: 

M=  0.4342944819  .    .    .;     Ioge10  =  1/Af  =  2.3025850930.    .    . 
0.4343  logez;  loge  x  =  2.3026  Iogi0z. 

For  tables  of  multiples  of  M  and  1/Af,  see  p.  62.     For  graphs  of  the  logar- 
ithmic and  exponential  functions,  see  p.  174;  series,  p.  160. 

The  Binomial  Theorem.  (For  table  of  binomial  coefficients,  see  p. 
39  and  p.  116.) 

_        .   N  •  n(n  -  1)  n(n  -  l)(n  -  2) 

Let  (n)l  -  n,  (n),  -  -—  -*     (»),  -        1X2X3       • 

=  --- 


1X2X3X4 
Then,  for  any  value  of  n,  provided  1  x  \  <  1, 

(1  +  z)n  =  1  +  (n)ix  +  (n)2z2  +  (n)3*3  +  (n)4*4  +  .    ..   . 
(If  n  is  a  positive  integer,  the  series  breaks  off  with  the  term  in  xn,  and  is 
valid  without  restrictions  on  x,  see  p.  112.) 

The  most  useful  special  cases  are  the  following: 


(bl  <  i) 


(1  +  o:)3  =  (1  +  xY*   =  1  + 


128 

with  corresponding  formulae  for  \/l  —  x,  etc.,  obtained  by  reversing    the 
signs  of  the  odd  powers  of  x.     Also,  provided  |6|  <  |a|: 

(a  +  6)n  =  an  ( 1 

where  (n)i,  (71)2,  etc.,  have  the  values  given  above. 

Arithmetical  Progression.  In  an  arithmetical  progression,  a;  a  +  d\ 
a  +  2d]  a  +  3d;  .  .  .,  each  term  is  obtained  from  the  preceding  term 
by  adding  a  constant,  called  the  constant  difference,  d.  If  n  is  the  number  of 
terms,  the  last  term  is  I  =  a  -j-  (n  —  l)d;  the  "average"  term  is  H(«  +  0{ 


FORMAL  ALGEBRA  115 


and  the  sum  of  the  n  terms  is  n  times  the  average  term,  or  S  =  ftn(a  +  Z). 
The  arithmetical  mean  between  a  and  b  is  (a  +  b)/2. 

Geometrical  Progression.  In  a  geometrical  progression,  o;  or;  or2; 
or8  ;  .  .  .  ,  each  term  is  obtained  from  the  preceding  term  by  multiplying  by  a 
constant,  called  the  constant  ratio,  r.  The  nth  term  is  arn~l.  The  sum 
of  the  first  n  terms  is  S  =  a(rn  -  l)/(r  -  1)  =  a(l  -  rn)/(l  -  r).  If 
r  is  a  positive  or  negative  fraction,  that  is,  if  —  1  <  r  <  +1,  then  rn  will 
approach  zero  as  n  increases,  and  the  sum  of  n  terms  will  approach  o/(l  —  r) 
as  a  limit.  The  geometric  mean  betweep  a  and  6  is\/~ab;  also  called  the 
mean  proportional  between  a  and  6  (p.  113;  construction,  p.  102). 

The  harmonic  mean  between  a  and  6  is  2ab/(a  +  b). 

Summation  of  Certain  Series  by  Second  and  Third  Differences. 
Let  01,  a2,  as,  .    .    .  On  be  any  series  of  n  numbers,  as  in  the 
first  column  of  the  adjoining  scheme.     By  subtracting      » 
each  number  from  the  next  following,  form  the  column     Jo      d       fd 
of  "first  differences,"  and  by  repeating  this  process,  form 
the  columns  of  second,  third,   etc.,  differences.     If  the     fc       «       la       •« 
kth  differences  are  all  equal,  so  that  subsequent  differ-  M       & 

ences  are  all  zero,  the  original  series  is  called  an  arithme-   "27    37     —  is 
tical  series  of  the  fcth  order.     In   this  special  case  the   _  g    1y     -12 
series  can  be  summed  as  follows:  Denote  the  numbers   —   1      i     ~  6      Q 
which  stand  at  the  head  of  the  successive  columns  of        9      1          9      6 
differences  by  />',£>",  D"',  ....      Then  the  nth  term  of        8      7  ' 

the  series  is  an,  and  the  sum  of  the  first  n  terms  is  Sn, 
where 


(n  - 

1  X  & 

(n  -  l)(n  -2)(n  - 


n(n  -  1)     ,  '    n(n-l)(n-2) 

D 


1X2X3 

*>n» 

1X2X3 


n(n-l)(n-2)(n-3)      „, 
1X2X3X4 

If  the  series  is,  for  example,  of  the  third  order,  each  of  these  formulae 
will  stop  with  the  term  involving  £>'";  and  only  a  few  terms  of  the  series  are 
required  for  the  computation  of  the  D's.  (Differentials,  p.  159.) 

Sum  of  the  Squares  or  Cubes  of  the  First  n  Natural  Numbers. 
1+2+3  +  .    .    .  +  (n  -  1)  +  n  =  %n(n  +  1). 

12  +  22  +  32  +  .    .    .  +  (n  -  I)2  +  n2  =  fcn(n  +  l)(2n  +  1). 

13  +  23  +  33  +  .    .    .   +  (n  -  I)3  +  n3  =  [j*n(n  +  I)]2. 
Formula  for  Interpolation  by  Second  Differences.     In  any  ordinary 

table  giving  a  quantity  y  as  a  function  of  a  variable  x,  let  it  be  required  to 
find  the  value  of  y  corresponding  to  a  value  of  x  which  is  not  given  directly 
in  the  table,  but  which  lies  between  two  tabulated  values,  as  x\  and  xz.  If 
x  =  xi  +  md,  where  d  =  xz  —  x\  =  the  constant  interval  between  two  suc- 
cessive x's,  and  m  is  some  proper  fraction,  then  the  corresponding  value  of 
y  will  be  given  by  the  formula 

m(m  -  1)  m(m  -  l)(m  -  2)  • 

y  =  Vl  +  mD    +       1X2     D     +  -       :  x  2<x  3        D      +  .    .    . 

where  D',  D",  D'",   .        .   are  the  first,  second,  third,  .    „    .  differences  in  the 


116 


ALGEBRA 


series  of  j/'s  which  begins  with  y\  (see  above),  provided  the  function  is  oi 
such  a  nature  that  the  differences  of  higher  orders  become  negligibly  small. 
The  coefficients  of  D',  D",  D"',  ...  in  the  formula  are  the  binomial  coeffi- 
cients for  fractional  values  of  m  (see  following  table).  The  several  terms  of 
the  formula  (with  careful  attention  to  sign)  are  the  successive  corrections 
which  must  be  added  to  y\\  the  sum  of  these  corrections  should  be  rounded 
out  to  the  nearest  unit  of  the  last  significant  place  before  adding.  If  D' 
<  4,  the  term  involving  D",  and  later  terms,  can  be  neglected;  the  formula 
then  reduces  to  y  =  y\  +  mD',  which  is  the  familiar  formula  for  ordinary, 
or  "linear,"  interpolation.  If  Z>'"  <  8  (or  D"  "  <  12,  or  D"  '"  <  16),  the 
term  involving  D'"  (or  D"  ",  or  D"  '")  can  be  neglected. 

Binomial  Coefficients  for  Fractional  Values  of  m 


m 

(m). 

(m), 

(m)4 

(m)s 

0.0 

-  0.0000 

0.0000 

-0  0000 

00000 

O.I 

-  0.0450 

0.0285 

-  0.0207 

0.0161 

0.2 

-  0.0800 

0.0480 

-  0  0336 

0  0255 

0.3 

-0.1050 

0  0595 

-  0  0402 

0  0297 

0.4 

-  0.1200 

0.0640 

-  0.0416 

0  0300 

0.5 

-  0.1250 

0.0625 

-  0.0391 

0.0273 

0.6 

-  0.1200 

0  0560 

-  0.0336 

0.0228 

0.7 

-  0.1050 

0  0455 

-  0.0262 

0.0173 

0.8 

-  0.0800 

0.0320 

-  0.0176 

0  0113 

0.9 

-  0.0450 

0.0165 

.  -  0.0087 

0.0054 

„      - 
Here 


m(m  - 


,    N 
(«). 


w(m-l)(m-2)(m-3) 


'  6tC' 


1X2  1X2X3  1X2X3X4 

Compare  p.  39. 

Permutations.  The  number  of  possible  permutations  or  arrangements 
of  n  different  elements  is  1  X  2  X  3  X  .  .  .  X  n  =  n\  (read:"n  factorial"). 

If  among  the  n  elements  there  are  p  equal  ones  of  one  sort,  q  equal  ones 
of  another  sort,  r  equal  ones  of  a  third  sort,  etc.,  then  the  number  of  possible 
permutations  is  (nl)/(p\  X  q\  X  rl  X  .  .  .),  where  p  +  g  +  r  +  .  .  .  =  n. 

Combinations.  The  number  of  possible  combinations  or  groups  of 
n  elements  taken  r  at  a  time  (without  repetition  of  any  element  within  any 
one  group),  is  [n(n  -  l)(n  -  2)(w  -  3)  .  .  .  (n  -  r  +  l)]/(r!)  ==  (n)r. 
(See  table  of  binomial  coefficients,  p.  39.)  If  repetitions  are  allowed,  so 
that  a  group,  for  example,  may  contain  as  many  as  r  equal  elements,  then 
the  number  of  combinations  of  n  elements  taken  r  at  a  time  is  (m)r,  where 
m  =  n  +  r  -  1.  NOTE:  (n)i  +  (n)a  +  .  .  .  +  (n)«  =  2»  -  1. 

SOLUTION  OF  EQUATIONS  IN  ONE  UNKNOWN  QUANTITY 

Roots  of  an  Equation.  An  equation  containing  a  single  variable  x 
will  in  general  be  true  for  some  values  of  x  and  false  for  other  values.  Any 
value  of  x  for  which  the  equation  is  true  is  called  a  root  of  the  equation. 
To  "solve"  an  equation  means  to  find  all  its  roots.  Any  root  of  an  equation, 
when  substituted  therein  for  x,  will  "satisfy"  the  equation.  An  equation 
which  is  true  for  all  values  of  x,  like  (x  +  I)2  =  xz  +  2x  +1,  is  called  an 
identity  [often  written  (x  +  I)2  s  x*  +  2x  +  1]. 

Types  of  Equations. 

(a)  Algebraic  Equations: 

of  the  first  degree  (linear),  e.gr.,   2x  +  6  =0   (root:  x  =  —3); 
of  the  second  degree    (quadratic),    e.g.,    xz  —  2x  —  3  =0  (roots:    —  1,  3); 
of  the  third  degree  (cubic),  e.g.,  x3  -  6x2  +  5x  +  12  =0  (roots:  -  1,  3,  4). 


SOLUTION  OF  EQUATIONS  IN  ONE  UNKNOWN  QUANTITY       117 

(6)  Transcendental    Equations: 

exponential  equations,  e.g.,  2X  =  32  (root:  x  =  5);  2X  =  —  32  (no  root); 
trigonometric  equations,  e.g.,  10  sin  x  —  sin  3x  =  4  (roots:  30°,  150°). 

Legitimate  Operations  on  Equations.  An  equation  which  is  true  for 
a  particular  value  of  x  will  remain  true  for  that  value  of  x  after  any  one  of 
the  following  operations  is  performed: 

Adding  any  quantity  to  both  sides;  subtracting  any  quantity  from  both 
sides;  transposing  any  term  from  one  side  to  the  other,  provided  its  sign 
be  changed;  multiplying  or  dividing  both  sides  by  any  quantity  which  is 
not  zero;  changing  the  signs  of  all  the  terms;  raising  both  sides  to  any  positive 
integral  power;  extracting  any  odd  root  of  both  sides;  extracting  any  even 
root  of  both  sides,  provided  the  ±  sign  is  used;  taking  the  logarithms 
of  both  sides  (both  sides  being  positive)  ;  taking  the  sin,  cos,  tan,  etc.,  of  both 
sides. 

Notice,  however,  that  the  new  equation  obtained  by  some  of  these  operations  may 
possess  "additional  roots"  which  did  not  belong  to  the  original  equation.  This  occurs 
especially  when  both  sides  are  squared;  thus,  x  =  —2  has  only  one  root,  namely,  —  2; 
but  z2  =-  4,  obtained  by  squaring,  has  not  only  the  root  —  2  but  also  another  root,  +  2. 

Equations  of  the  First  Degree  (Linear  Equations).  Solution:  Collect 
all  the  terms  involving  x  on  one  side  of  the  equation,  thus:  ax  —  b,  where 
a  and  b  are  known  numbers.  Then  divide  through  by  the  coefficient  of  x, 
obtaining  x  =  b/a  as  the  root. 

Equations  of  the  Second  Degree  (Quadratic  Equations).  Solution: 
Throw  the  equation  into  the  standard  form  ax2  +  bx  +  c  =  0.  Then  the 
two  roots  are:  _ 

-  6  +  Vb*  -  4ac  -b  -Vb*  -4ac 

Xl=~       -2aT  **  =  -^a~ 

The  roots  are  real-and-distinct,  coincident,  or  imaginary,  according  as 
62  —  4ac  is  positive,  zero,  or  negative.  The  sum  of  the  roots  is  x\  +  xt 
=  —  b/a]  the  product  of  the  roots  is  x\xz  =  c/a. 

GRAPHICAL  SOLUTION.  Write  the  equation  in  the  form  z2  =  px  +  q,  and  plot  the 
parabola  yi  =  x2,  and  the  straight  line  yi  =  px  +  q.  The  abscissae  of  the  points  of 
intersection  will  be  the  roots  of  the  equation.  If  the  line  does  not  cut  the  parabola, 
the  roots  are  imaginary. 

Equations  of  the  Third  Degree  with  Term  in  x2  Absent.  Solution: 
After  dividing  through  by  the  coefficient  of  x3fc  any  equation  of  this  type 
can  be  written  re3  =  Ax  +  B.  Letp  =  A/3  and  q  =  B/2.  The  general  solu- 
tion is  as  follows: 

Case  1.     q2  —  p3  positive.     One  root  is  real,  namely 

zj     =   %/q    +   Vq^  p3    +  ^/q    -   Vq*~^  P3; 
the  other  two  roofs  are  imaginary. 

Case  2.     g2  —  p3  =  zero.     Three  roots  real,  but  two  of  them  equal. 


Case  3.     g2  —  p5  negative.     All  three  roots  real  and  distinct.     Determine 
an  angle  u  between  0  and  180°,  such  that  cos  u  =  q/(pV^p).     Then 
xi  =  2Vpcos  (w/3),  xz  =  2^/p  cos  (w/3  +  120°),  xz  =  2\/pcos  (w/3  +  240°). 

GRAPHICAL  SOLUTION.     Plot  the  curve  yi  =  x3,  and  the   straight  line  yt  =  Ax  +  B. 
The  abscissae  of  the  points  of  intersection  will  be  the  roots  of  the  equation. 

Equations  of  the  Third  Degree  (General  Case).     Solution:  The  gen- 
eral'cubic  equation,  after  dividing  through  by  the  coefficient  of  the  highest 


118  ALGEBRA  ,. 

power,  may  be  written  rr3  +  ax*  -f  bx  -f  c  =  0.  To  get  rid  of  the  term  in 
a;2,  let  x  =  xi  —  a/3.  The  equation  then  becomes  #i3  =  Ax\  +  B,  where 
A  =  3(a/3)2  -  6,  and  B  =  -  2(a/3)3  +  6(a/3)  -  c.  Solve  this  equation 
for  xi,  by  the  method  above,  and  then  find  x  itself  from  x  =  xi  —  (a/3). 

GRAPHICAL  SOLUTION.  Without  getting  rid  of  the  term  in  x2,  write  the  equation  in 
the  form  x3  =  -  a[x  +  (&/2a)p  +  [a(6/2a)2  -  c],  and  solve  by  the  graphical  method. 

General  Properties  of  Algebraic  Equations.  An  algebraic  equation  of 
the  nth  degree  in  x  is  an  equation  of  the  type 

aoxn  +  aixn~l  +  oax71"2  +  .   .   .  -fan-iz+  an  =  0 

where  the  a's  are  any  given  numbers  (ao  not  zero),  the  expression  on  the 
left  being  called  a  polynomial  of  the  nth  degree  in  x.  Such  an  equation 
will,  in  general,  have  n  roots;  but  some  of  these  n  roots  may  be  equal,  and 
some  may  be  imaginary.  Imaginary  roots  always  occur  in  pairs. 

If  the  equation  is  written  in  the  form:  (a  polynomial  in  x)  =  0,  then  (1) 
if  a  is  a  root  of  the  equation,  x  —  a  is  a  factor  of  the  polynomial;  (2)  if  the 
polynomial  can  be  factored  in  the  form  (x  —  p)(x  —  q)(x  —  r)  .  .  .  =  0, 
each  of  the  quantities  p,  q,  r,  .  .  .  is  a  root  of  the  equation;  (3)  if  x  is  very 
large  (either  positive  or  negative) ,  the  higher  powers  of  x  are  the  most  impor- 
tant; (4)  if  x  is  very  small,  the  higher  powers  may  be  neglected. 

Short  Method  of  Substitution  in  a  Polynomial.  To  find  the  value  of 
4x«  —  14z3  +  23x  —  26  when  x  =  3,  for  example,  first  arrange  the  terms 
in  order  of  descending  powers  of  x,  and  write  the  detached  coefficients,  with 
their  signs,  in  a  row,  taking  care  to  supply 

a  zero  coefficient  for  any  missing  term,  in-  4      —  14         0        23      —  26  (3 
eluding  the  constant  term.     Then,  beginning  12—6—18  15 

at  the  left,  bring  down  the  first  coefficient;  -     

multiply  this  by  3,  and  add  to  the  second  4       —2—6  5—11 

coefficient;  multiply  this  result  by  3  again, 

and  add  to  the  third  coefficient;  and  so  on.  The  final  result,  —  11,  is  the 
value  of  the  polynomial  when  x  =  3. 

Short  Method  of  Dividing  a  Polynomial  by  x  —  a.  The  device  just 
explained  gives  not  only  the  value  of  the  polynomial  when  x  =  3,  but  also  the 
result  of  dividing  the  polynomial  by  x  —  3.  Thus,  in  the  case  illustrated, 
the  quotient  is  4o;3  —  2x2  —  Qx  +  5  and  the  remainder  is  —  11.  That  is, 
4z«  -  14z3  +  Ox2  +  23x  -  26  =  (x  -  3)(4z3  -  2z2  -  6x  +  5)  -  11. 

Exponential  Equations.  To  solve  an  equation  of  the  form  a*  =  6, 
take  the  logarithms  of  both  sides:  x  log  a  =  log  6,  whence  x  =  (log  6) /(log  a). 
For  example,  if  3*  =  0.4,  x  =  log  0.4/log  3  =  (0.6021  -  1)/0.4771  = 
—  0.3979/0.4771  =  —  0.8340.  Notice  that  the  complete  logarithm  must  be 
taken,  not  merely  the  mantissa. 

Trigonometric  Equations.  (1)  To  solve  a  cos  x  +  6  sin  x  =  c,  where 
a  and  b  are  positive:  Find  the  acute  angle  u  for  which  tan  u  =  b/a,  and  the 
angle  v  (between  0  and  180°)  for  which  cos  v  =  c/\/az  +  62.  Then  xi  =  u  +  v 
and  xz  =  u  —  v  are  roots  of  the  equation. 

(2)  To  solve  a  cos  x  —  b  sin  x  =  c,  where  a  and  6  are  positive:  Find  u 
and  v  as  above.  Then  x\  =  —  (u  +  v)  and  xz  =  —  (u  —  v)  are  roots  of  the 
equation. 

General  Method  of  Solution  by  Trial  and  Error.  This  method  is 
applicable  to  a  numerical  equation  of  any  form,  and  can  be  carried  out  to 
any  desired  degree  of  approximation.  It  is  especially  useful  when  a  first 
approximation  to  a  root  is  already  known,  Write  the  equation  i^  the  form 


SOLUTION  OF  SIMULTANEOUS  EQUATIONS  119 

f(x)  =  0,  where  /(x)  means  any  function  of  x,  and  plot  the  curve  y  =  f(x)  for 
a  sufficient  number  of  values  of  x  to  obtain  a  general  idea  of  the  shape  of  the 
curve.  Then  pick  out  the  regions  in  which  the  curve  appears  to  cross  the 
axis  of  x,  and  plot  the  curve  more  accurately  in  each  of  these  regions.  Thus, 
<by  successive  approximations,  plotting  the  important  parts  of  the  curve  on  a 
larger  and  larger  scale,  determine  as  accurately  as  necessary  the  points  where 
the  curve  crosses  the  axis — that  is,  the  values  of  x  which  make/(z)  equal  to  zero. 
Thus,  suppose  that  f(x)  =  3.0  when  x  =  2.6  and  —  5.0  when  x  =  2.7  (see  Fig.  1). 
Then  the  curve  must  cross  the  axis  somewhere  between  x  =  2.6  and  x  =  2.7;  and  since 
it  will  not  yary  greatly  from  a  straight  line  between  those  points,  it  is  seen  that  it  must 


-^,0.7 


•;w^ 


2.6 


2.64 


2.63 


-S.O 


FIG.   1. 

cross  near  2.64.  Suppose  the  value  of /(x)  when  computed  for  x  =  2.64,  is  —  0.2,  and 
when  computed  for  x  =  2.63  is  +  0.7;  then  the  root  lies  between  x  -  2.63  and  2.64. 
Plotting  this  section  on  the  larger  scale,  it  is  seen  that  the  next  guess  should  be  about 
2.638;  and  so  on. 

Instead  of  writing  the  original  equation  with  all  the  terms  on  the  left-hand  side,  it  is 
often  better  to  divide  the  expression  into  two  parts,  say  fi(x)  and/Kx),  writing  the  equa- 
tion in  the  form/i(x)  =  MX).  If  then  the  two  curves  j/i  =  /i(x)  and  3/2  =  MX)  be  plotted 
separately,  on  the  same  diagram,  the  value  of  x  corresponding  to  their  point  of  inter- 
section will  be  the  desired  root. 

SOLUTION  OF  SIMULTANEOUS  EQUATIONS 

The  Meaning  of  a  System  of  Simultaneous  Equations.  To  solve  a 
system  of  n  simultaneous  equations  in  n  unknowns,  means  to  find  all  the  sets 
of  values  of  the  unknowns  (if  any)  which,  when  substituted  in  the  given 
equations,  will  satisfy  all  the  equations  at  the  same  time.  If  a  system  of 
equations  has  no  solution,  the  equations  are  "inconsistent;"  if  it  has  an  in- 
finite number  of  solutions,  the  equations  are  "not  all  independent." 

Simultaneous  Eguations  of  the  First  Degree  in  Two  Unknowns. 
Factors 


(1)   aix  +  b\y  =  c\  \       62 


(2)  azx  +  b2y  =  c2      - 


—  az 


(aibz  —  azbi)x  =  bzci  —  biCz  .".  x  =  (bzci  —  biCz)  /((lib*  —  azbi) 
(aibz  —  azb\)y  =  a\cz  —  azci  .'.  y  =  (a\cz  —  azci)  / (aibz  —  azbi) 
Here  (1)  is  multiplied  by  62,  (2)  by  —  61,  and  the  products  added  so  as  to 
eliminate  y,  again,  (1)  is  multiplied  by  —  az,  (2)  by  01,  and  the  products 
added  so  as  to  eliminate  x.  (The  process  is  most  conveniently  performed  as 
follows:  Write  the  multipliers,  as  62  and  —  61,  at  the  right  of  the  equations; 
multiply  the  first  term  of  each  equation  by  its  proper  multiplier  and  add; 
then  multiply  the  second  term  of  each  equation  by  its  proper  multiplier,  and 
add;  and  so  on.  This  is  simpler  than  the  common  practice  of  multiplying 
out  each  equation  separately  before  adding.)  If  0162  —  azbi  =  0,  the  equa- 
tions have  no  solution  when  c\  7^  cz,  and  an  infinite  number  of  solutions  when 


120 


ALGEBRA 


GI  =  C2.     The  following  special  solution  is  possible  when  the  sum 
difference  of  the  two  unknowns  are  given: 

Let  x  +  y  =  m        (1) 

and  x  —  y  =  n        (2) 
(1)  +  (2) 


and 


2):         2x  =  m  +  n         .m.x  = 
(1)   -  (2):         2y  =  m  -  n         .'.  y  =  W(m  -  n) 
Simultaneous  Equations  of  the  Second  Degree  in  Two  Unknowns, 

(a)   When  the  product  of  the  unknowns,  and  their  sum  or  difference,  are  given: 


*  +y  =  5 
xy  =  4 

Squaring  (1),  x* 
From  (2), 

+  2xy  +  2/2  = 
-4*2/ 

25 
-  16 

Adding,            *2 

—  2*2/  +  y2  = 

9 

Hence,              x 
But                   * 

—  y  =  3  or  —  3 
+  y  =  5  or       5 

Therefore        * 

y 

=  4          *  =  1 
=  1     °r  y  =  4 

(1) 

(2) 

x  —  y  =  3 
*2/  =  4 

(1) 
(2) 

*2  -  2*y  +  2/2 
4xy 

=     9 
=  16 

x*  +  2*2/  +  2/2 

=  25 

x  +  2/  =  5  or  - 
x  —  y  =  3  or 

-  5 
3 

x  =  4           re  = 
IT-  1     °r  y  = 

-  1 

-  4 

(b)   When  the  product  and  the  sum  of  the  squares  are  given: 

xy  =    5  (1)   V(4)  :  x  +  y  =  6  or       6  or—  6  or—  6 


26  (2) 


*-2/=4or-4or       4  or  -4 


From  (1), 

2*2/  =  10  (3) 

.".  * 

=  5 

I 

_  i 

-  5 

(2) 

+  (3):  *2  _|_2*2/ 

+  2/2  =  36  (4) 

•'•y 

=  1 

or        _ 
o 

or        ^ 

or 

-  1 

(2) 

-(3): 

r2  — 

2*2/ 

+  y2  =  16  (5) 

(C) 

When 

the  sum 

or  difference,  and  the  sum  of 

the 

squares,  are  given: 

*    + 

y 

=    5     (1) 

*    - 

-    y 

= 

3 

(D 

*'  + 

2/2 

=  17     (2) 

*2     _|_         y, 

17 

(2) 

(Ds: 

*2    + 

2*2/ 

+  2/2   =25 

(D2: 

*2  - 

-  2xy  +  2/2  = 

9 

(2)  : 

*2 

+  2/2  =  17 

(2)  : 

*2                         +   I/2     =    17 

(I)2  -  (2)  : 

2*2/ 

=    8 

(D2  - 

(2)  : 

-  2a-y 

= 

-8 

xy  =4 

Then  proceed  as  in  case  (a),  above. 


xy  =4 

Then  proceed  as  in  case  (a),  above. 


(d)  When  one  equation  is  of  the  first  degree  and  the  other  of  the  second,  as 
ax  +  by  =  c,  and  Ax2  +  Bxy  +  Cy*  +  Dx  +  Ey  +  F  =0:  Solve  the  first 
equation  for  y  in  terms  of  x,  and  substitute  in  the  second.  This  will  give  a 
quadratic  equation  in  x.  Solve  this  quadratic  for  the  frwo  values  of  x,  and 
for  each  of  these  values  of  x  find  the  corresponding  value  of  y  by  substituting 
in  the  equation  of  the  first  degree. 

Simultaneous  Equations  of  the  First  Degree  in  n  Unknowns.  For 
example:  Factors 


(a) 

(c) 
(d) 

2x 
5x 
3x 
4x 

-  y 

-  4y 

+  3z  +  5w  =  29 
-  2z  +  3«;  =  15 
+  72  -   w  =  12 
-  5«  +  2w  =  3 

3 
-  5 

1 
5 

2 
-  5 

(0) 

-  19* 

-2ly 
-  17y 

+  192  = 

+  382 

+  312  = 

=  12 
=  89 
=  43 

-  2 

1 

-  31 
19 

(h) 

55* 
285* 

+  5y 
+  802/ 

=  65 
=  445 

16 
-1 

METHOD  OF  LEAST  SQUARES  121 

(;)       595*  =595;  .  /.  x  =  1; 

5y  =    65  -  55*  =  65     -  55  =  10;  .'.  y  =  2; 

192  =     12  +  19*  +  13y  =  12  +  19  +  26  =  57;  .'.z   =3; 

2w  =      3  -    4*  -    3y  +  5z  =  3  -  4  -  6  +  15  =  8;  .'.  w  =  4. 

Here  tu  is  eliminated  from  (a)  and  (6) ,  obtaining  (e) ;  from  (a)  and  (c) , 
obtaining  (/) ;  and  from  (a)  and  (d),  obtaining  (g).  Then  z  is  eliminated  from 
(e)  and  (/),  obtaining  (A),  and  from  (e)  and  (0),  obtaining  (i).  Then  $/  is 
eliminated  from  (h)  and  (i),  obtaining  0'),  which  contains  only  the  single  vari- 
able *.  Hence  *  =  1.  Now  substituting  this  value  of  *  in  either  (h)  or  (t), 
y  is  found;  substituting  these  values  of  *  and  y  in  either  (e),  (/),  or  (g),  z  is 
found;  and  so  on.  (Solution  by  determinants,  see  p.  123.) 

Approximate  Solution  of  a  Set  of  Simultaneous  Equations  of  the 
First  Degree  When  the  Number  of  Equations  is  Greater  Than  the 
Number  of  Unknowns.  (Method  of  Least  Squares.) 

Case  1.  Single  Unknown  Quantity.  Given  n  equations  in  one  un- 
known *;  for  example,  n  equally  careful,  independent  measurements  of  some 
physical  quantity: 

X    —  Xl,  X    =  *2,     .      .      .     *    =  Xn- 

As  the  "best"  value  of  *,  take  the  arithmetic  mean,  XQ,  of  the  several  deter- 
minations, namely,  XQ  =  (*i  +  xz  +  .  .  .  +  *n)  /n.  The  quantities  vi  = 
*o  —  xi,  z>2  =  XQ  —  *2,  .  .  .  vn  =  XQ  —  xn  are  called  the  residuals  of  the 
observed  values  with  respect  to  XQ,  and  their  absolute  values  (that  is,  their 
numerical  values  without  regard  to  sign)  are  denoted  by  \vi\,  \v^\,  .  .  .  |t>n|. 
[It  can  be  shown  that  the  sum  of  the  squares  of  the  residuals  with  respect 
to  XQ  is  smaller  than  the  sum  of  the  squares  of  the  residuals  with  respect  to 
any  other  value  X'Q-  hence  the  name  of  the  method:  "least  squares."] 
The  quantities  r  and  TQ,  defined  exactly  by  Bessel's  formulae: 

0.6745 


n(n  -  1) 
or  given  approximately  by  the  simpler  formulae  of  Peters: 

°'8453     (M  +  H  +  .      .  +  W), 


are  called  the  probable  error  of  a  single  observation  (r),  and  the  probable 
error  of  the  mean  (T-Q),  for  the  given  series  of  observations.  Note  that 
ro  =  r l\/n.  For  tables  of  the  coefficients,  see  p.  63.  This  quantity  r  (or 
ro)  is  best  regarded  as  merely  a  conventional  means  of  recording  the  relative 
precision  of  different  sets  of  observations.  If  r  is  small,  it  may  be  inferred 
that  most  errors  of  the  "  accidental"  class  have  been  eliminated;  but  it  should 
be  especially  noted  that  the  smallness  of  r  gives  no  information  in  regard  to 
"constant"  or  "systematic"  errors. 

A  statement  like  ''x  is  equal  to  2.36  with  a  probable  error  of  0.02,"  is 
written:  x  =  2.36  +  0.02,  and  is  usually  understood  to  mean  that  the  true 
value  of  x,  as  far  as  can  be  told,  is  just  as  likely  to  lie  inside  as  outside  the 
interval  from  2.34  to  2.38. 


122  ALGEBRA 

'  To  test  the  distribution  of  residuals,  arrange  the  residuals  in  order  of 
magnitude,  without  regard  to  sign,  and  count  the  number,  y,  of  residuals 
which  are  numerically  less  than  some  assigned  value  a;  divide  y  by  w,  the 
total  number  of  observations,  and  divide  a  by  r,  the  probable  error  of  a  single 
observation.  Do  this  for  various  values  of  a,  and  compare  the  results  with 
the  table  on  p.  63,  which  gives  the  standard  distribution  of  residuals,  as 
found  from  experience  from  a  large  number  of  different  series  of  observations. 
In  particular,  the  number  of  residuals  numerically  less  than  r  should  be  about 
equal  to  the  number  numerically  greater  than  r  (if  n  is  large).  If  any  large 
discrepancy  appears,  the  series  of  observations  should  be  regarded  as  unsatis- 
factory. 

NOTE.  The  "mean  square  error"  sometimes  met  with  is  equal  to  the  probable  error 
divided  by  0.6745. 

Case  2.     Several  Unknown  Quantities.     Assume  that  there  have  been 

obtained  by  measurement  or  observation  n  different  equations  of  the  first 

degree   involving,    say,   three  unknown  quantities, 

Given  Equations  x,  y,  z.     There  are  then  n  simultaneous  equations 

a\x  +  biy  +  c\z  =  pi      in  three  unknowns,  and  if  n  >  3  there  will  be,  in 

azx  +  b%y   +  czz  =  pz      general,  no  set  of  values  of  x,  y,  z  which  will  satisfy 

all    these    n    equations   exactly.       In  such  a  case, 

anx  -f-  bny  +  CnZ  =  Pn  the  "best"  set  of  values,  XQ,  yo,  ZQ,  may  be  found 
by  the  method  of  least  squares  as  follows.  (The 

process  usually  involves  a  large  amount  of  labor;  the  use  of  a  computing 
machine  is  advisable.) 

First,  arrange  the  n  given  equations  in  the  form  indicated,  being  careful 
not  to  modify  any  of  them  by  multiplication  or  division.  (Any  of  the  coeffi- 
cients may  of  course  be  zero.) 

Next,  form  the  three  "normal  equations"  as  follows:  (1)  •  Multiply  each 
of  the  given  equations  by  the  coefficient  of  x  in  that  equation,  and  add;  the 

result  will  be  the  first  normal  equation. 

Normal  Equations  (2)  Multiply  each  of  the  given  equations 

[aa]a;o  +  [ab]yo  +  [ac]zo  =  [ap]  by  the  coefficient  of  y  in  that  equation,  and 
[ba]xo  -\-  [bb]yo  +  [bc]zo  =  [bp]  add;  the  result  will  be  the  second  normal  equa- 
[ca]xo  +  [cb]yo  +  [cc]zo  =  [cp]  tion.  (3)  Similarly  for  the  third.  {  Nota- 
tion: [aa]  =  ai2  +  a22  +  .  .  .  +  an2; 
[ab]  =  ai&i  +0262  -f-  .  .  .  +anbn',  [ap]  =  aipi  -\-aipz  +  .  .  .  +anpn:  etc.} 

Finally,  solve  the  three  normal  equations  for  the  three  unknowns  in  the 
usual  way. 

The  quantities  DI  =  0,1X3  +  &i2/o  +  C\ZQ  —  pi,  etc.,  are  called  the  residuals 
with  respect  to  XQ,  yo,  ZQ.  [It  can  be  shown  that  the  sum  of  the  squares  of  the 
residuals  with  respect  to  XQ,  yo,  ZQ  is  smaller  than  the  corresponding  quantity 
with  respect  to  any  other  set  of  values,  x'o,  y'o,  z'o;  this  relation  is  taken  as  the 
criterion  for  the  "best"  set  of  values  of  x,  y,  z.] 

The  probable  error  of  a  single  observation  is 

0.6745 


+  vz2  +  •    •    •   +  *>n2,  or  approximately, 

(hi  +  M  +  .    .    .   +  hi), 


n(n  —m) 
where  m  =  the  number  of  unknown  quantities  (here  m  =  3). 


DETERMINANTS 


123 


DETERMINANTS 

Determinants  are  used  chiefly  in  formulating  theoretical  results;  they  are 
seldom  of  use  in  numerical  computation. 


Evaluation  of  Determinants  : 
Of  the  second  order: 


|a2&2| 


0162  — 


Of  the  third  order: 


02&2C2 


=   Ol 


&2c2 


&3C3 


&1C1 

he 


—   &3C2)     —   02(6lC3    — 

Of  the  fourth  order: 

&2C2C?2 

,      —  *«•!  va^a**a    —  dz  baCada 
dabacada 


6404^ 


bzczdz 
baCadal 


etc.  In  general,  to  evaluate  a  determinant  of  the  nth  order,  take  the  ele- 
ments of  the  first  column  with  signs  alternately  plus  and  minus,  and  form  the 
sum  of  the  products  obtained  by  multiplying  each  of  these  elements  by  its 
corresponding  minor.  The  minor  corresponding  to  any  element  ai  is  the 
determinant  (of  next  lower  order)  obtained  by  striking  out  from  the  given 
determinant  the  row  and  column  containing  01. 

Properties  of  Determinants. 

1.  The  columns  may  be  changed  to  rows  and  the  rows  to  columns: 

010203 


a262c2 

036303 


ClC2C3 


2.  Interchanging  two  columns  changes  the  sign  of  the  result. 

3.  If  two  columns  are  equal,  the  determinant  is  zero. 

4.  If  the  elements  of  one  column  are  m  times  the  elements  of  another 
column,  the  determinant  is  zero. 

5.  To  multiply  a  determinant  by  any  number  m,  multiply  all  the  elements 
of  any  one  column  by  m. 


7. 


Solution  of  Simultaneous  Equations  by  Determinants. 

If    aix  +  biy  +  ciz  =  pi 


01  +  PI  +  3i,     bi     ci 

ai&ici 

pibici 

01&lCl| 

a2  +  p2  +  0.2,     bz     Cz 

= 

02&2C2 

-|- 

Pzbzcz 

-f- 

qzbzczl 

03  +  Pa  +  qa,      ba     ca 

03&3C3 

pabaca 

qabacal 

dibiCi 

01  +  mbi,  &i     ci 

azbzcz 

= 

o2  +  mbz,  bz     Cz 

aabaCa 

03  +  mba,  ba     Ca 

aix  +  bzy  +  czz  =  p2 
aax  +  bay  +  caz  =  PS 
then   x  =  Di/D, 

y  =  D2/Z>,  where  Di 
z    =  Da/D, 


where  D 


PI&IC 


a2&2c2 
dabaca 


0, 


pabaca 


aapac 


Da  = 


Similarly  for  a  larger  (or  smaller)  number  of  equations. 


124 


ALGEBRA 


THE  ALGEBRA  OF  IMAGINARY  OR  COMPLEX  QUANTITIES 

In  the  algebra  of  imaginary  or  complex  quantities,  the  objects  on  which  the 
operations  of  the  algebra  are  performed  are  not  numbers  in  any  ordinary 
sense  of  the  word,  but  are  best  thought  of  as  points  in  a  plane  (or  as  vectors 
drawn  from  a  fixed  origin  to  these  points).  The  "complex  plane"  is  de- 
termined by  three  fundamental  points,  O,  U,  i,  arranged  as  in  Fig.  2  and  called 
the  zero  point,  the  unit  point,  and  the  imaginary  unit  point,  respectively. 
All  points  on  the  line  through  O  and  U  are  called  real  points — positive  if 
on  the  right  of  O,  negative  if  on  the  left.  All  the  remain- 
ing points  in  the  plane  are  called  imaginary  points — 
those  on  the  line  through  0  and  i  being  called  the  pure 
imaginary  points. 

The  position  of  any  point  A  in  the  plane  may  be  de- 
termined by  the  distance  from  the  origin  O,  measured  in 
terms  of  OU  as  the  unit  length,  and  the  angle  <p  which 
OA  makes  with  the  positive  direction  of  the  axis  of  reals. 
The  distance  r  is  sometimes  called  the  modulus  or  ab- 
solute value  of  the  point;  the  angle  <f>  is  sometimes  called 
the  amplitude  or  argument  of  the  point.  The  notation 


FIG.  2. 


A  =  (3,   ^120°) 

means  the  point  whose  distance,  r,  is  3  times  OU,  and  whose  angle,  <p,  is  120°. 
The  development  of  the  algebra  depends  wholly  on  the  definitions  of  three 
fundamental  operations  denoted  by  A  +  B,  A  X  B,  and  eA,  as  follows. 

Addition  and  Subtraction.     The  sum,   A  +  B,  of  two  points  A  and  B 
is  defined  as  the  point  reached  by  starting  from  A  and  performing  a  journey 
equal  in  length  and  direction  to  the  journey  from  0  to  B.     That  is,  the  vector 
from  O  to  A  +  B  is  the  vector  sum  of  the  vectors  OA  and 
OB.     In  case  A  and  B  are  not  in  line  with  0,  the  point  A  +  B        •  A+B 

is  the  fourth  vertex  of  a  parallelogram  of  which  OA  and  OB 
are  the  sides  (Fig.  3).  Conversely,  if  any  two  points  A  and 
B  are  given,  there  is  a  definite  point  X  such  that  A  =  B  +  X ; 
this  point  X  is  called  the  remainder,  A  minus  B,  and  is  FIG.  3. 

denoted  by  A  —  B.     The  point  O  —  B  is  denoted  for  brevity 
by  —  B.     With  these  definitions  of  A  +  B  and  A  —  B,  all  the  ordinary  laws 
of  addition  and  subtraction  that  hold  in  the  algebra  of  real  numbers  hold  also 
in  the  algebra  of  complex  quantities.     In  particular,  the  zero  point  O  has  all 
the  formal  properties  of  the  number  zero,  and  is  denoted  by  0. 

[Note:     If  A  and  B  are  "real"  points,  A  -f-  B  and  A  —  B  will  also  be  real. 

Repeated  Addition.  Multiples  and  Submultiples.  The  point 
A+A+A-f-.  .  .  +  A  to  n  terms  is  called  the  nth  nlultiple  of  A  and 
is  denoted  by  nA.  The  points  U,  2U,  3U,  .  .  .  are  denoted,  for  brevity, 
by  1,  2,  3,  .  .  ..  Conversely,  if  any  point  A,  and  any  positive  integer  n 
are  given,  there  is  a  definite  point  X  such  that  nX  =  A ;  this 
point  X  is  called  the  nth  submultiple  of  A,  and  is  denoted  by 
A/n.  The  points  £7/2,  U/3,  ...  are  denoted,  for  brevity, 
by  ^,  H,  .  .  .. 

Multiplication  and  Division.  The  product,  A  X  B,  or 
A-B,  or  AB,  of  two  points  A  and  B  is  defined  as  the  point 
whose  angle  is  the  sum  of  the  angles  of  the  given  points,  and 
whose  distance  is  the  product  of  the  distances.  (See  Fig.  4.) 
Thus,  if  A  =  (5,  ^120°)  and  B  =  (2,^270°),  then  AB  = 
(10,  -^30°).  Conversely,  if  any  two  points  A  and  B  are  given, 
provided  B  is  not  zero,  there  is  a  definite  point  X  such  that 


THE  ALGEBRA  OF  IMAGINARY  OR  COMPLEX  QUANTITIES       125 


FlO.   5. 


A  =  BX.  This  point  X  is  called  the  quotient,  A  divided  by  B,  and  is  de- 
noted by  A/B  (where  B  ^  0).  Thus,  the  point  A/B  is  a  point  whose  angle 
is  the  angle  of  A  minus  the  angle  of  B,  and  whose  distance  is  the  distance  of 
A  divided  by  the  distance  of  B.  The  point  U/B  (B  7*  0)  is  called  the 
reciprocal  of  the  point.fi,  and  is  denoted  by  l/B.  (See  Fig.  5.)  With  these 
definitions  of  AB  and  A/B  the  elementary  laws  of  multiplication  and  division 
that  hold  in  the  algebra  of  real  numbers  hold  also  in  the  algebra  of  complex 
quantities.  In  particular,  the  point  U  has  all  the  formal 
properties  of  the  number  unity,  and  is  denoted  by  1. 

[Note:    If  A  and  B  are  real,  AB  and  A/B  will  also  be  real.] 

Repeated  Multiplication.     Powers  and  Roots.     The 
point  A  X  A  X  A  X    .    .    .    X  A  to  n  factors  is  called  the 
nth  power  of  A  and  is  denoted  by  An  (Fig.  6).     Conversely, 
if  any  point  A  (not  0)  and  any  positive  integer  n  are  given, 
there  will  be  n  distinct  points  X  such  that  Xn  =  A ;  each  of 
these  points  is  called  an  nth  root  of  A,  some  one  of  them, 
usually  the  one  with  the  smallest  positive  angle,  being  de- 
noted by   \/A  or  A1/n.      Thus,  the  point   \/A 
is  a  point  whose  distance  is  the  nth  root  of  the  dis- 
tance of  A,  and  whose  angle  is  I/nth  of  the  angle 
of  A.     All  the  nth  roots  of  A  will  lie  on  the  cir- 
cumference of  a  circle  about  O  as  center,  and  will 
divide  that  circumference  into  n  equal  parts  (Fig. 
7).     Every  point  A  (not  0)  has  two  square  roots, 
three  cube  roots,  etc.     Hence  the  theorem  "If  An 
=  Bn  then  A  =  B"  does  not  hold  in  this  algebra, 
and  the  ordinary  rules  for  radical  signs  must  be 
applied  with  caution.     For  example,  if  A  and  B 
are  positive  reals,  V  —  A-V  —  B  =  —  V ' AB  and 
not  V(  —  A)(  —  #),  which  would  give  +  \fAB. 

[Note:  If  A  is  real  and  positive,  \/A  will  be  real  and 
positive;  if  A  is  real  and  negative,  -\^A will  be  real  if  n 
is  odd  and  imaginary  if  n  is  even.] 

Properties  of  i.  The  point  i  is  the  point  whose  dis- 
tance is  1  and  whose  angle  is  90  deg.  It  follows  from 
the  definition  above  that  multiplying  any  point  A  by 
i  has  the  effect  of  rotating  the  point  through  an 
angle  of  +  90°  without  changing  its  distance  from  O. 
In  particular, 

i2  =  -  1,  i9  -  -  i,  i*  =  1,  i5  =  i,  etc.;  i  =  V-l, 
—  i  =  —  V  —  1;  where  "1"  denotes  not  the  number 
one,  but  the  point  U. 

Similarly,  multiplying  any  point  A  by  —  1  has  the 
effect  of  rotating  the  point  through  180  deg. 

First  Standard  Form  for  a  Complex  Quantity 
(Fig.  8).  Any  point  A  can  be  expressed  in  the  form 
x  +  iy,  where  x  and  y  are  real  points.  For  example, 
the  three  cube  roots  of  1  are  1,  —  #  +  ^i\^3,  and 


j-2/ 


FIG.  8. 


-H— 


126  ALGEBRA 

In  general,  (x\  +  iyi)  +  (xz  +  iyd  =  (x\  +  xz)  +  i(yi  +  1/2)  ; 

(xi  +  iyi)(xt  +  ij/j)   =  (xix*  -  2/13/2)  +  i(x*yi  +  zii/2)  | 

xi  +  iyi  _  xixz  +  3/13/2    _,    .  333/1  —  xiyz 
xz  +  iyz         xz*  +  3/22  xj  +  3/22 

If  two  complex  quantities  are  equal,  their  real  parts  must  be  equal,  and  the 
coefficients  of  their  pure  imaginary  parts  must  also  be  equal.  That  is,  if 
x\  +  iyi  =  xz  +  iyz,  then  xi  =  x%  and  y\  =*  2/2.  Thus  a  single  equation  between 
complex  quantities  is  equivalent  to  two  equations  between  real  quantities. 

Conjugate  Imaginaries.  Two  points  A  =  x  +  iy  and  B  =  x  —  iy  are 
called  conjugate  imaginaries.  Two  such  points  are  symmetrically  situated 
with  regard  to  the  axis  of  reals.  The  sum  and  product  of  two  conjugate 
imaginaries  will  be  real. 

Second  Standard  Form  for  a  Complex  Quantity.  Since  x  •=  r  cos  <p  and 
y  —  r  sin  <f>,  any  point  A  =*  x  +  iy  can  be  expressed  A  =  r  (cos  <p  -f-  i  sin  <f>), 
where  r  is  real  and  positive  (namely,  the  distance  of  A),  and  <p  is  real 
(namely  the  angle  of  A).  For  example,  the  three  cube  roots  of  1  are  1, 
cos  120°  +  i  sin  120°,  and  cos  240°  +  i  sin  240°.  In  general, 
[n  (cos  <f>i+i  sin  <f>i)]  [r2(cos  </>2+i  sin  <p2)]  =nr2[(coa  (<pi+<f>t)  +*  sin  (^1+^2)]; 
[r(cos^  +  iam<p)]n  =  rn[coa(n<p)  +  ism  (n<p)]  (De  Moivre's  Theorem). 

The  Exponential  Function,  eA,  or  exp  A,  of  any  point  A  =x  +  iyis  defined 
aa  the  point  whose  distance  is  ex  and  whose  angle  (measured  in  radians)  is  y. 
That  is,  .ex+iv  =  e*(cos  y  +  i  sin  y).  Here  ex  means  the  ordinary  expo- 
nential function  of  the  real  quantity  x,  where  e  =  2.718. 

From  this  definition,  the  usual  formal  laws  of  exponents  can  be  deduced: 

6AeB    =  eA+Bf     (eA)n    =  gn4>    Q-A    =  \/eA.    e\    =  e>    eO    =   lt 

The  function  eAis  a  periodic  function  with  a  pure  imaginary  period  2iri; 
that  is,  e  A±k'iiri  =  eAt  where  k  is  any  positive  integer. 

If  A  is  made  to  move  along  a  line  parallel  to  the  axis  of  reals  [or  axis  of  pure 
imaginaries],  the  corresponding  point  eA  will  move  along  a  straight  line  through 
0  [or  along  a  circle  about  O  as  center]. 

Properties  of  elv>.  The  point  el(p  is  a  point  whose  distance  is  1  and  whose 
angle  is  <f>.  It  follows  from  the  definitions  above  that  multiplying  any 
point  A  by  el<f>  has  the  effect  of  rotating  the  point  through  an 
angle  <p,  without  changing  its  distance  from  O.  In  particular,  elir  —  —  1, 
e~iv  =  -  1;  e™/z  =  i;  e~i7r/2  =  -  i;  e™  =  1. 

Third  Standard  Form  for  a  Complex  Quantity.  Any  point  A  can  be 
expressed  in  the  form  A  =  rel<f>>  where  r  is  the  distance  and  <p  the  angle  of  the 
point.  For  example,  the  three  cube  roots  of  1  are  1,  e*27™,  e'27r*.  In  general, 


If  x  +  iy  =  rel<f>,  then  r  =  Vz2  +  y2,  sin  *  =  —  ,  cos*>  =  —  ,  tan  <?  =  — 

If  two  complex  quantities  are  equal,  their  distances  will  be  equal,  and  their 
angles  will  differ  at  most  by  some  multiple  of  2,-ir.  Thus,  if  nel<f>1  =  rzel<f* 
then  ri  =  r-i  and  <p\  =  <pz  or  <p2  ±  k2ir.  Here  again  a  single  equation  between 
complex  quantities  is  equivalent  to  two  equations  between  real  quantities. 


THE  ALGEBRA  OF  IMAGINARY  OR  COMPLEX  QUANTITIES       127 

Definition  of  AB.    Let  A  =  reW;  then  AB  =  exp  [(loge  r  +  i<p)B]. 

For  example,  i{  =  e-V2  where  i  =  \/  —  1. 

If  a  is  a  positive  real,  a  x      y  =  ax  [cos  (j/  loge  a)  +  i  sin  (y  log«  a)]. 

Trigonometric  and  Hyperbolic  Functions  of  a  Complex  Variable. 

If   A  is  any  point,   then,  by  definition, 

eiA    _  e-iA  eiA    +  e-iA  sin   A 

sin  A   = : ,    cos  A  = — ,    tan  A  =  —  (cos  A  /^  0) ; 

2  cos  A 


.  „          „  eA  +  e  A  smh 

smh  A  =  -  -  -  ,  cosh  A  =  -  -  --  ,  tanh  A 


,  ,  — 

2  2  cosh  A 

Hence  the  formulae  that  hold  for  these  functions  in  the  real   case   (p.   131; 
p.  135;  p.  161)  hold  also  for  the  complex  case.     Further: 

sin  (a;  -h  iy)     =  sin  x  cosh  y    +  i  cos  x  sinh  y,          sin  iy    =  i  sinh  y  ; 

cos  (x  +  iV)    =  cos  x  cosh  y    —  i  sin  a;  sinh  j/,          cos  ty    =  cosh  y; 

sinh  (aj  +  iy)   =  sinh  a;  cos  y    +  i  cosh  a:  sin  j/,  sinh  t'y  =  i  sin  y; 

cosh  (x  +  *2/)   =  cosh  x  cos  ?/    +  i  sinh  x  sin  y,          cosh  iy  =  cos  y; 
where  sin  x,  sinh  x,  etc.,  are  the  ordinary  trigonometric  and  hyperbolic  func- 
tions of  the  real  variables  x  and  y.     The  functions  sin  A  and  cos  A  are  periodic 
with  a  real  period  2,-tr.     The  functions  sinh  A  and  cosh  A  are  periodic  with  a 
pure  imaginary  period  "2iri. 

Logarithmic  and  Other  Inverse  Functions  of  a  Complex  Variable. 
If  any  point  A  is  given,  there  will  be  an  infinite  number  of  points  X  such 
that  ex  =  A;  any  one  of  these  points  may  be  called  a  logarithm  of  A,  and 
be  denoted  by  log  A.  All  the  values  of  the  logarithm  of  A  may  be  obtained 
from  any  one  value  by  adding  multiples  of  2iri. 

If  x  -f-  iy  =  rel<f>,  then  loge  (x  +  iy)   =  loge  r  +  i<p  +  k-Zin. 

If  any  point  A  is  given,  there  will  be  an  infinite  number  of  points  X  such  that 
sin  X  =  A;  any  one  of  these  may  be  denoted  by  sin-1  A.  The  functions 
cos-1  A,  sinh-1  A,  etc.,  are  defined  in  a  similar  way. 

The  elementary  laws  of  operation  which  hold  for  these  functions  in  the 
algebra  of  reals  hold  also,  in  a  general  way,  in  the  algebra  of  complex  quanti- 
ties; but  caution  must  be  used,  on  account  of  the  ambiguity  in  the  symbols 
log  A,  sin  -^A,  etc.,  which  denote  many-valued  functions. 

Differentiation  of  Functions  of  a  Complex  Variable.  If  w  =/(z), 
_  the  derivative  of  w  with  respect  to  z  is  defined  as 

dw/dz  =  lim  {  [f(z  -f-  Az)  —  f(z)]/Az]  when  Az  approaches  0. 

It  can  be  shown  that  lim  {[exp  Az  —  l]/Az}  =  1;  hence  d(e?)  =  e*dz, 
d(sin  z)  =  cos  z  dz,  etc.,  so  that  the  formulae  for  differentiation  here  are  the 
same  as  in  the  case  of  a  real  variable  (p.  157). 


NOTE.     For  the  algebra  of  vector  analysis,  which  differs  in  important  respects  from 
the  algebra  of  complex  quantities,  see  p.  185. 


TRIGONOMETRY 


FIG.  2. 


FORMAL  TRIGONOMETRY 

Angles,  or  Rotations.  An  angle  is  generated  by  the  rotation  of  a  ray, 
as  Ox,  about  a  fixed  point  O  in  the  plane.  Every  angle  has  an  initial  line 
(OA)  from  which  the  rotation  started  (Fig.  1),  and  a  terminal  line  (OB) 
where  it  stopped;  and  the  counterclockwise  direction  of  rotation  is  taken  as 
positive.  Since  the  rotating  ray  may  revolve  as  often  as 
desired,  angles  of  any  magnitude,  positive  or  negative, 
may  be  obtained.  Two  angles  are  congruent  if  they 
may  be  superposed  so  that  their  initial  lines  coincide  and 
their  terminal  lines  coincide.  That  is,  two  congruent 
angles  are  either  equal  or  differ  by  some  multiple  of  360 
deg.  Two  angles  are  complementary  if  their  sum  is  90 
deg. ;  supplementary  if  their  sum  is  180  deg.  (The  acute 
angles  of  a  right-angled  triangle  are  complementary.)  If 
the  initial  line  is  placed  so  that  it  runs  horizontally  to  the 
right,  as  in  Fig.  2,  then  the  angle  is  said  to  be  an  angle  in 
the  1st,  2nd,  3rd,  or  4th  quadrant  according  as  the 
terminal  line  lies  across  the  region  marked  I,  II,  III,  or  IV. 
The  angles  0  deg.,  90  deg.,  180  deg.,  270  deg.  are  called  the 
quadrantal  angles. 

Units  of  Angular  Measurement. 

(1)  SEXAGESIMAL  MEASURE.     (360  degrees  =  1  revolution.)     1  degree  = 
1°  =  Ho  of  a  right  angle.     The  degree  is  usually  divided  into  60  equal  parts 
called  minutes  (0,  and  each  minute  into  60  equal  parts  called  seconds  ("); 
while  the  second  is  subdivided  decimally.     But  for  many  purposes  it  is  more 
convenient  to  divide  the  degree  itself  into  decimal  parts,  thus  avoiding  the  use 
of  minutes  and  seconds.     (See  tables,  pp.  46-51.) 

(2)  CENTESIMAL    MEASURE,    used    chiefly    in    France.     (400   grades  =  1 
revolution.)      1  grade  =  Moo  of  a  right  angle.     The  grade  is  always  divided 
decimally,  the  following  terms  being  sometimes  used:  1  "centesimal  minute" 

=  Moo  of  a  grade;  1  "centesimal  second"  =  Moo  of  a  centesimal  minute.  In 
reading  Continental  books  it  is  important  to  notice  carefully  which  systerh 
is  employed. 

(3)  RADIAN,  OR  CIRCULAR,  MEASURE.     (IT  radians  =  180  degrees.)     1  radian 
=  the  angle  subtended  by  an  arc  whose  length  is  equal  to  the  length  of  the 

radius.  The  radian  is  constantly  used  in  higher  mathematics  and  in-  me- 
chanics, and  is  always  divided  decimally.  Table,  pp.  44^45. 

1  radian  =  57°.30-   =  57°.2957795131   =  57°  17' 44".806247  =  180° ,V. 

1°  =  0.01745  .  .  .  radian  =  0.0174532925  radian. 

1'  =  0.00029  08882  radian.     1"  =  0.00000  48481  radian. 

(For   10-place  conversion    tables,    see   the   Smithsonian 
Tables  of  Hyperbolic  Functions,  Washington,  D.  C.) 

Definitions  of  the  Trigonometric  Functions.     Let 
x  be  any   angle  whose  initial  line  is  OA  and  terminal  line 
OP  (see  Fig.  3).     Drop  a  perpendicular  from  P  on  OA  or 
OA  produced.     In  the  right  triangle  OMP,  the  three  sides 
are  MP    =    "side  opposite"  O  (positive  if  running  upward);  OM  =  "side 
adjacent"  to  O  (positive  if  running  to  the  right);  OP  =  " hypothenuse "  or 
"radius"  (may  always  be  taken  as  positive);   and  the  six  ratios  between 
these  sides  are  the  principal  trigonometric  functions  of  the  angle  x;  thus: 

128 


FORMAL  TRIGONOMETRY 


129 


sine  of  £  =  sin  re  =  opp/hyp  =  MP/OP; 
cosine  of  x  =  cosx  =  adj/hyp    =  OM/OP; 
tangent  of  x  =  tana;  =  opp/adj    =  MP/OM; 
cotangent  of  x  =  cot  x  =  adj/opp    =  OM/MP; 
secant    of   x  =  sec  x  =  hyp/adj    =  OP/OM; 
cosecant    of    x  =  cso  x  =  hyp/opp  =  OP/MP. 
The  last  three  are  best  remembered  as  the  reciprocals  of  the  first  three: 

cot  x  =  I/tan  x]  sec  x  =  I/cos  x;  esc  x  =  I/sin  x. 

Other  functions  in  use  are  the  versed  sine,  the  coversed  sine,  and  the  ex- 
terior secant: 

vers  x  =  1  —  cos  x;  covers  x  =  1  —  sin  x;  exsec  x  =  sec  x  —  1. 
For  graphs,  see  p.  174;  series,  p.  161. 

Signs  of  the  Trigonometric  Functions 


If  x  is  in  quadrant 

I 

II 

III 

IV 

sin  x  and  esc  x  are       .        .... 

-f 

+ 

cos  x  and  sec  x  are  

+ 



+ 

tan  x  and  cot  x  are  

+ 

-f 

vers  x  and  covers  x  are  always  positive. 

Variations  in  the  Functions  as  z  Varies  from  0  deg.  to  360  deg.  are 

shown  in  the  accompanying  table.     The  variations  in  the  sine  and  cosine  are 


FIG. 


best  remembered  by  noting  the  changes  in  the  lines  MP  and  OM  (Fig.  4) 
in  the  "unit  circle"  (that  is,  a  circle  with  radius  =  OP  =  1),  asP  moves  around 
the  circumference. 


X 

0°  to  90° 

90°  to  180° 

180°  to  270° 

270°  to  360° 

Values  at 

30° 

45° 

60° 

sin  x 

CSC  X 

+0to+l 

+  00  tO+1 

+1to+0 

+HO  +  OO 

-Oto-l 

—  oo  tO—  1 

-lto-0 

-ItO-oo 

M 

2 

W2 
V2 

H\/3 

^Vs 

cos  x 

sec  x 

+  Ito+0 
+1  to  +  oo 

-Oto-l 

-cotO-1 

-lto-0 

—  ItO—  oo 

+0to+l 

+  ootO+l 

H\/3 
*S\/3 

H\/2 
V2 

H 

2 

tan  x 

cot  * 

+0to+oo 

+  ootO+0 

-  oo  tO  -0 
-OtO-co 

+0  tO  +  <x> 
+  on  tO  +0 

-  oo  tO  -0 

-Oto-  °° 

H-S/3 
V3 

1  1 
1 

V3 
H-S/3 

vers  x 
covers  x 

+0to+l 
+lto+0 

+lto+2 
+0to+l 

+2to+l 

+  lto+2 

+HO+O 

+2to+l 

\/2  =1.4142;  H\/2  =0.7071;  A/3  =1.7321;  ^VV  =  0.8660;  W3  =0.5774;  ^A/3  =1.1547 

Trigonometrical  Tables.  The  tables  on  pp.  46-56  give  the  values  of  the 
principal  trigonometric  functions  and  of  their  logarithms,  correct  to  four 
places  of  decimals,  the  angle  advancing  either  by  tenths  of  a  degree  (p.  46) 
or  by  10  min.  (p.  52).  These  tables  will  be  found  adequate  for  most 


130 


TRIGONOMETRY 


computations  in  which  an  accuracy  of  1  part  in  1000  is  sufficient.  If  much 
computing  is  to  be  done,  it  is  advisable  to  use  a  separate  volume  of  tables, 
containing  more  facilities  for  interpolation,  and  printed  in  larger  type, 
such  as  the  four-place  tables  of  E.  V.  Huntington  (Harvard  Cooperative 
Society,  Cambridge,  Mass.),  with  convenient  marginal  tabs;  the  five-place 
tables  published  by  Macmillan  or  many  others;  the  six-place  tables  of 
Bremiker;  the  standard  seven-place  tables  of  Schron,  Vega,  or  Bruhns 
(angles  advancing  by  10  sec.);  or  the  great  eight-place  of  Bauschinger 
and  Peters  (angles  advancing  at  intervals  of  1  sec.  fromO  deg.  to  90  deg.). 
The  larger  tables  give  only  the  logarithms  of  the  functions,  not  the  natural 
values. 

To  Find  Any  Function  of  a  Given  Angle.  (Reduction  to  the  first 
quadrant.)  It  is  often  required  to  find  the  functions  of  any  angle  x  from  a 
table  that  includes  only  angles  between  0  deg.  and  90  deg.  If  x  is  not 
already  between  0  deg.  and  360  deg.,  first  "reduce  to  the  first  revolution"  by 
simply  adding  or  subtracting  the  proper  multiple  of  360  deg. ;  [for  any  func- 
tion of  (z)  =  the  same  function  of  (x  ±  n  X  360°)].  Next  reduce  to  the 
first  quadrant  as  follows: 


If  x  is  between 

90°  and  180° 

180°  and  270° 

270°  and  360° 

Subtract 

90°  from  x 

180°  from  x 

270°  from  x 

Then     sin  x 

=  +cos  (x-90°) 
=  +sec  (a;  -90°) 
=  -sin  (x-90°) 
*=-csc  (x-90°) 
=  -cot  (x-90°) 
=  -tan  (x-90°) 

=  -sin  (x-180°) 
=  -csc  (a:  -180°) 
=  -cos  (x-180°) 
=  -sec  (x  -180°) 
=  +tan  (x-180°) 
=  +cot  (s-1800) 

=  -cos  (x-270°) 
=  -sec  (x  -270°) 
=  +sin  (x-270°) 
=  +csc  (x  -270°) 
=  -  cot  (x  -  27C°; 
=  -tan  (x  -270°) 

esc  x  

COS  X 

sec  x  

tan  x 

cot  x  

vers  x  

=  l+sin  (x  -90°) 
=  l-cos  (a;  -90°) 

=  l+cos  (a-  180°) 
=  l+sin    (a;  -180°) 

=  l-sin  (z-27<n 
=  l  +  cos(a;-2700) 

covers  x  

The  "reduced  angle"  (x  —  90°,  or  x  —  180°,  or  x  —  270°)  will  in  each  case 
be  an  angle  between  0°  and  90°,  whose  functions  can  then  be  found  in  the 
table. 

[NOTE.    The  formulae  for  sine  and  cosine  are  best  remembered  by  aid  of  the  unit  circle.] 
To  Find  the  Angle  When  One  of  Its  Functions  is  Given.     In  general, 
there  will  be  two  angles  between  0  deg.  and  360  deg.  corresponding  to  any  given 
function.     The  following  tabulated  rules  show  how  to  find  these  angles. 


Given 

First  find  from  the  tables 
an  acute  angle  xo  such  that 

Then  the  required  angles  xi 
and  xz  will  be 

sin  3=  -fa 
cos  x=  -\-a 
tan*=  +o 
cot  x=  +o 

sin  xo  =  a 
cos  xo  =  a 
tan  xo  =  a 
cot  xo  =  a 

xo        and    180°  -xo 
xo        and  [3  60°  -so] 
xo        and  [180°+a;o] 
xo       and  [180°  +  xo] 

sin  x=  —  a 
cos  x  =  —  a 
tan  x  =  —  o 
cot  x=  —  a 

sin  xo  =  a 
cos  xo  =  a 

tan  xo  =  a 
cot  xo  =  a 

[180°  +  a:o]and 
180°  —  xo  and 
180°  -xo  and 
180°  -xo  and 

360°-a:o 
180°  +  xo 
360°  -xo 
360°  -xo 

The  angles  enclosed  in  brackets  lie  outside  the  range  from  0  deg.  to  180  deg.,  and  hence 
cannot  occur  as  angles  in  a  triangle. 

For  solution  of  trigonometric  equations,  see  p.  118. 


FORMAL  TRIGONOMETRY 


131 


Relations  Between  the  Functions  of  a  Single  Angle.     (See  Fig.  5.) 
sin   x  1  cos  x 


cos  x 


V 1  —  sin2  x 


'1  +  tan2x       VI  +  cot2x 
Functions  of  Negative  Angles,     sin  (— x)  =  —  sin 

cos  (  — x)   =  cos  x;  tan  (  — x)   =  —  tan  x. 

Functions  of  the  Sum  and  Difference  of  Two  Angles. 

sin  (x  +  y)   =  sin  x  cos  y  +  cos «  sin  y; 

cos  (x  +  2/)   =  cos  x  cos  y  —  sin  x  sin  y; 

tan  (x  -f  j/)  =  [tan  x  +  tan  y]/[l  —  tan  x  tan  j/]; 

cot  (x  +  y)   =  [cot  x  cot  y  —  l]/[cot  x  +  cot  y\\ 

sin  (x  —  y)   =  sin  x  cos  y  —  cos  x  sin  y; 

cos  (x  —  y}   —  cos  x  cos  y  +  sin  x  sin  y\ 

tan  (x  —  y)   =  [tan  x  —  tan  2/]/[l  +  tan  x  tan  j/]; 

cot  (x  —  y)  =  [cot  xcoty  +  l]/[cot  #  —  cot  x]; 

sin  x  +  sin  y  =  2  sin  ^(x  +  y)  cos    ^(x  —  j/) ; 

sin  x  —  sin  y  =  2   cos   ^(x  +  y)    sin   >£(x  —  y); 

cos  x  +  cos  y  =  2  cos   y*(x  +  j/)  cos   ^(x  —  y}\ 

cos  x  —  cos  y  =  —  2  sin  W(x  +  #)  sin  \i(x  —  y); 

sin  (x  +  y)  sin  (x  +  y) 

tan  &  -f  tan  y  =  -  — ;    cot  x  +  cot  j/  =  — 

cos  x  cos  y  sin  x  sin  |/ 


FIG.  5. 


sin  (x  —  y) 


cot  x  —  cot  y 


—  x) 


tan  x  —  tan  y  = 

cos  x  cos  j/  sin  x  sin  y 

sin2 x  —  sin2  2/  =  cos2  y  —  cos2  x  =  sin  (x  +  y)  sin  (x  —  y); 

cos2  x  —  sin2  y  =  cos2  |/  —  sin2  x  =  cos  (x  +  y)  cos  (x  —  y) ; 

sin   (45°  +  x)   =  cos  (45°  -  x) ;   tan  (45°  +  x)  =  cot  (45°  -  x) ; 

sin  (45°  -  x)  =  cos  (45°  +  x):   tan  (45°  -  x)   =  cot  (45°  +  x). 
In  the  following  transformations,  a  and  b  are  supposed  to  be  positive, 
c  =  V^2  +  b2,  A  =  the  positive  acute    angle  for  which  tan  A  =  a/b,  and 
B  =  the  positive  acute  angle  for  which  tan  B  =  b/a: 

a.cos  x  +  b  sin  x  =  c  sin  (A  +  x)    =  c  cos  (B  —  x) ; 

a  cos  x  —  6  sin  x  =  c  sin  (A  —  x)   =  c  cos  (B  +  x). 

Functions  of  Multiple  Angles  and  Half  Angles. 

sin  2x  =  2  sin  x  cos  x;  sin  x  =  2  sin   MX  cos  ^x; 
cos  2x  =  cos2  x  —  sin2  x  =  1  —  2  sin2  x  =  2  cos2  x  —  1 ; 
2  tan  x  cot2  x  —  1 


tan2x 


1  -  tan2  x' 


cot  2x 


2  cot  x 

3  tan  x  —  tan3  x 
sin  3x  =  3  sin  x  —  4sm3x;    tan  3x  = 


cos  3x  =  4  cos3  x  —  3  cosx; 


1-3  tan2  x 


132  TRIGONOMETRY 

sin  (n*)  =  n  sin  x  cos  w~1  x  —  (n)a  sin3  *  cosn~3* 

+  (n)s  sin5  a;  cos  n~6 

cos  (n*)  «=  cosn  a;  —  (n)z  sin2  a;  cos  n-2*  +  (n) 4  sin4  a;  cos  n~4  a 
where  (n)2,  (n)8,  .    .    .   are  the  binomial  coefficients  (see  p.  39). 
sin  M  x  =  ±  \A$(1  —  cos  a;);  1  —  cos  x  =2  sin2  MX; 


cos  MX  =  +  V^(l  +  cos  *);  1  -f-  cos*  =  2  cos2 

,_  —  cos  *  sin  *  1  —  cos  * 

tan  . _ 

cos  *  sin  * 

/*  \  /I    4-  sin   -r 

tan   (a  +45C 


—  sin  * 

Here  the+  or  —  sign  is  to  be  used  according  to  the  sign  of  the  left-hand 
side  of  the  equation. 

Relations  Between  Three  Angles  Whose  Sum  is  180°.  , 

sin  A    +  sin  B  +  sin  C  =  4  cos  MA  cos  MB  cos  MC; 

cos  A    +  cos  B  +  cos  C  =  4  sin  MA  sin  MB  sin  MC  +  1 ; 

sin  A    -\-  sin  B  —  sin  C  =  4  sin  MA  sin  J&B  cos  MC; 

cos  A    -j-  cos  B  —  cos  C  =  4  cos  ^ A  cos  MB  sin  }£C  —  1 ; 

sin2  A  +  sin2J5  +  sin2  C  =  2  cos  A   cos  £  cos  C  +  2; 

sin2  A  +  sin2 B  —  sin2  C  =  2  sin  A  sin  B  cos  C; 

tan  A   +  tan  B  +  tan  C  =  tan  A  tan  B  tan  (7; 

cot^A+  cot  MB  +  cot  ^(7  =  cot  MA  cot  £1B  cot  MC; 

cot  A  cot  B  +  cot  A  cot  (7  +  cot  B  cot  (7  =  1; 

sin  2A  +  sin  2B  +  sin  2(7  =  4  sin  A  sin  B  sin  C; 

sin  2A  -j-  sin  2B  —  sin  2C  =4  cos  A  cos  B  sin  C. 

Inverse  Trigonometric  Functions.  The  notation  sin"1*  (read:  anti- 
sine  of  *,  or  inverse  sine  of  *;  sometimes  written  arc  sin  *)  means  the  prin- 
cipal angle  whose  sine  is  *.  Similarly  for  cos"1*,  tan"1*,  etc.  (The  prin- 
cipal angle  means  an  angle  between  —90°  and  +90°  in  case  of  sin"1  and 
tan"1,  and  between  0°  and  180°  in  the  case  of  cos"1.)  For  graphs,  see  p.  174. 

SOLUTION  OF  PLANE  TRIANGLES 

The  "parts"  of  a  plane  triangle  are  its  three  sides,  a,  6,  c,  and  its  three 
angles  A,  B,  C  (A  being  opposite  a,  B  opposite  b,  C  opposite  c,  and  A  + 
B  +  (7  =  180°).  A  triangle  is,  in  general,  determined  by  any  three  parts 
(not  all  angles).  To  "solve"  a  triangle  means  to  find  the  unknown  parts 
from  the  known.  The  fundamental  formula?  are: 

Law  of  sines :  -  =  —  — .      Law  of  cosines :  c2  =  a2  +  b2  —  2ab  cos  C. 
b       smB 

Right  Triangles.  Use  the  definitions  of  the  trigonom- 
etric functions,  selecting  for  each  unknown  part  a  relation 
which  connects  that  unknown  with  known  quantities;  then 
solve  the  resulting  equations.  Thus,  in  Fig.  6,  if  C  =  90°, 
then  A  +  B  -  90°,  c2  =  a2  +  62, 

sin   A  =  a/c,  cos  A  =  6/c,  tan  A  =  a/b,  cot  A  =  b/a. 

If  A  is  very  small,  use  tan  MA  =  \/(c—  6)/(c  +  6). 

Oblique  Triangles.  There  are  four  cases.  It  is  highly  desirable  in  all 
these  cases  to  draw  a  sketch  of  the  triangle  approximately  to  scale  before 
commencing  the  computation,  so  that  any  large  numerical  error  may  be 
readily  detected. 

Case  1.     GIVEN  Two  ANGLES  (provided  their  sum  is  <  180  deg.),  AND  ONE 


SOLUTION  OF  PLANE  AND  SPHERICAL  TRIANGLES 


133 


SIDE  (say  o,  Fig.  7).     The  third  angle  is  known,  since  A 


To  find  the  remaining  sides,  use  b 


a  sin  B 


a  sin  C 


+  B  +  C 

A 


=  180°. 


sin  A  sin  A 

Or,  drop  a  perpendicular  from  either  B  or  C  on  the  opposite 
side,  and  solve  by  right  triangles.  -pIQ    7 

Check:  c  cos  B  +  6  cos  C  =  a. 

Case  2.  GIVEN  Two  SIDES  (say  a  and  6),  AND  THE  INCLUDED  ANGLE 
(C) ;  and  suppose  a  >  6.  Fig.  8. 

First  Method:  Find  c  from  c2  =  a2  +  62  —  2ab  cos  C  [or  c2  =  (o  —  6)2  + 
2a&  vers  C];  then  find  the  smaller  angle,  5,  from  sin  B  =  (6/c)  sin  C;  and 
finally,  find  A  from  A  =  180°  —  (B  +  C).  Check:  a  cos  B  +  b  cos  A  =  c. 

Second  Method:     Find  ££(A  —  B)  from  the  law  of  tangents: 

tad  #(A  -  J3)  =  [(a  -  6) /(a  +  6)]  cot  &C, 
and  tf(A  +  B)  from  #(A  +  B)  =  90°—  C/2;  hence  A  = 
H(A  +  £)  +  #(A  -5)  and  B  =  y*(A  +  B)  -\<i(A  -  B). 
Then  find  c  from  c  =  o  sin  C/sin  A  or  c  =  6  sin  C/sin  B.  jrIQi  g. 

Check:  o  cos  B  +  6  cos  A  =  c. 

Third  Method:  Drop  a  perpendicular  from  A  to  the  opposite  side,  and 
solve  by  right  triangles. 

Case  3.  GIVEN  THE  THREE  SIDES  (provided  the  largest  is  less  than  the 
sum  of  the  other  two),  Fig.  9. 

First  Method:  Find  the  largest  angle  A  (which  may  be  acute  or  obtuse) 
from  cos  A  =  (62  +  c2  -  a2)/26c  {or  vers  A  =  [a2  -  (6  -  c)2]/26c}  and 
then  find  B  and  C  (which  will  always  be  acute)  from  sin  B  =  b  sin  A/o  and 
sin  C  =  c  sin  A/a.  Check:  A  +  B  +  C  =  180°. 

Second  Method:  Find  A,  B,  and  C  from  tan  \<iA  =  r/(s  —  a), 

tan  \<iB  =  r/(s  —  6),  tan  tfC  =  r/(s  —  c),  where  s  =  ^(a  +  6  +  c),  and 

+  B  +C 


r  =V(s  -  a)(s  -  6)(s  -  c)/s.     Check:     A 
Third  Method:  If  only  one  angle,  say  A,  is  required,  use 
sin   WA  =   A/(s  —  b)(s  —  c)/bc  or 


180°. 


cos    l$A  =  -vs(s  —  a) /be, 
according  as  ^A  is  nearer  0°  or  nearer  90°. 

Case  4.     GIVEN  Two  SIDES  (say  6  and  c)  AND  THE  ANGLE  FIG.  9. 

OPPOSITE  ONE  OF  THEM   (B).     This  is  the  "ambiguous 
case"  in  which  there  may  be  two  solutions,  or  one,  or  none  (see  Fig.  10). 

B  acute 


c*    c/ 

'NoSoIution)     (OneSolution)     (TwoSolutions) 


C, 


(One  Solution)  (One  Solution) 


B  obtuse 


»« 


-v 


(NoSolution)      (No5olution)       (NoSoMion)       (No  Solution) 

FIG.  10. 


(One  Solution) 


First,  try  to  find  C  from  sin  C  =  c  sin  B/b.  If  sin  C>  1,  there  is  no  solution. 
If  sin  C  =  1,  C  =  90°  and  the  triangle  is  a  right  triangle.  If  sin  C<  1, 
this  determines  two  angles  C,  namely,  an  acute  angle  Ci,  and  an  obtuse  angle 
Cj  =  180°  —  Ci.  Then  Ci  will  yield  a  solution  when  and  only  when 


134  TRIGONOMETRY 

Ci  +B  <  180°  (see  Case  1);  and  similarly  C2  will  yield  a  solution  when 

and  only  when  Cz  +  B  <  180°  (see  Case  1). 

Other  Properties  of  Triangles.     (See  also  p.  99  and  p.  105.) 

Area  =  tyab  sin  C  =  \/s(s  —  a)  (s  —  b)(s  —  c)  =  rs,  where  s  =  3,i(a  +  b  -f-  c), 

and  r  =radius  of  inscribed  circle  =  \/(s  —  a)(s  —  6)(s  —  c)/s. 
Radius  of  circumscribed  circle  =  R,  where 

2R  =  a/sin  A  =  6  /sin  B  =  c/sin  C;  r  =  4R  sin  4  sin  ^  sin  ?  =  ^-. 

222         4/ts 

The  length  of  the  bisector  of  the  angle  C  is 

-  c)        Vab[(a  +  b)2  -  c2] 


a  +b  a  +  6 


The  median  from  C  to  the  middle  point  of  c  is  m  =  }S\/2(a2  +  62)  —  c2. 

SOLUTION  OF  SPHERICAL  TRIANGLES 

For  the  occasional  solution  of  a  spherical  triangle  the  following  formulae 
will  be  sufficient.  For  a  detailed  discussion,  see  any  text-book  on  spherical 
trigonometry. 

Let  a,  b,  c  be  the  eides-of  the  spherical  triangle,  that  is,  portions  of  arcs  of 
great  circles  of  the  sphere;  and  let  A,  B,  C  be  the  angles  of  the  triangle,  that  is, 
the  angles  made  by  tangents  drawn  to  the  sides  at  their  points  of  intersection 
on  the  sphere.  The  sum  of  the  angles  will  always  be  greater  than  two  right 
angles,  and  may  be  nearly  six  right  angles.  The  angle  E  =  A  +  B  +  C  — 
180°  is  called  the  spherical  excess  of  the  triangle.  (See  also  p.  100.) 

sin  a        sin  b  sin  6        sin  c  sin  c        sin  a 

sin  A       sin  B  '         sin  B       sin  C1         sin  C       sin  A 

cos  a  =  cos  6  cos  c  +  sin  b  sin  c  cos  A, 
with  similar  formulae  for  cos  b  and  cos  c. 

cos  A  =  —  cos  B  cos  C  +  sin  B  sin  C  cos  a, 
with  similar  formulae  for  cos  B  and  cos  C. 

In  the  special  case  of  a  right  spherical  triangle,  in  which  C  =  90°, 

cos  A  cos  B 

cos  c  =  cos  a  cos  o  =cotA  cotB;   cos  a  =   — — — -  ;    cos  o  =    — — -; 

sin  B  sin  A 

sin  a  tan  b  tan  a 

sin  A  =  — ;    cos  A  =  ;    tan  A  =  — — —  • 

sin  c  tan  c  sin  o 

The  area  of  a  spherical  triangle  _  spherical  excess 
area  of  a  great  circle  180°. 


HYPERBOLIC  FUNCTIONS  135 


HYPERBOLIC    FUNCTIONS 

The  hyperbolic  sine,  hyperbolic  cosine,  etc.,  of  any  number  x,  are 
functions  of  x  which  are  closely  related  to  the  exponential  ex,  and  which  have 
formal  properties  very  similar  to  those  of  the  trigonometric  functions,  sine, 
cosine,  etc.  Their  definitions  and  fundamental  properties  are  as  follows 
(see  also  p.  127;  graphs,  p.  175;  table,  p.  60;  series,  p.  161): 

sinh  x  =  }$(ex  —  e~x);  cosh  a;  =  li(ex  +  e~x');  tanh  a;  =  sinh  a; /cosh  Z; 

csch  x  =  1/sinh  x;  sech  x  =  1/cosh  x;  coth  x  =  1/tanh  x; 

cosh2  x  —  sinh2  x  =1;  1  —  tanh2  x  =  sech2  x;  1  —  coth2  x  =  —  csch2  x; 

sinh  (  —  x)  =  —  sinh  x\    cosh  (  —  x)   =  cosh  x;  tanh  (  —  x)   =   —  tanh  x\ 

sinh  {x  ±  !/)  =  sinh  x  cosh  y  ±  cosh  x  sinh  y; 

cosh  (x  +  y)  =  cosh  x  cosh  y  ±  sinh  x  sinh  y; 

tanh  (a;  ±  !/)   =  (tanh  x  ±   tanh  !/)/(!  ±   tanh  x  tanh  y); 

sinh  2x  =  2  sinh  x  cosh  x;  cosh  2x  =  cosh2- a;  +  sinh2  x ; 

tanh  2x  =  (2  tanh  x) /(I  +  tanh2  x) ; 

sinh^z  =\As   (cosh  a;  —  1);  cosh  ftx  =  V^cosh  a;  +  1); 

tanh  MX  =  (cosh  x  —  l)/(sinh  x)   =  (sinh  o;)/(cosh  x  +  1). 

The  inverse  hyperbolic  sine  of  y,  denoted  by  sinh""1!/,  is  the  number 
whose  hyperbolic  sine  is  y;  that  is,  the  notation  x  =  sinh"1!/  means 
sinh  x  =  y.  Similarly  for  cosh-1!/,  tanh-1!/,  etc.  These  functions  are  closely 
related  to  the  logarithmic  function,  and  are  especially  valuable  in  the  integral 
calculus.  For  graphs,  see  p.  175. 

sinh-^/a)   =  loge(y  +  Vy«  +  o2)  —  loge  a; 

cosh-l(y/a)   =  loge(y  +  VV2  —  a2)  —  loge  a; 

y  £f_l-7y  7/  1J    -\-   d 

tanh-1-     =  ^loge  -^-^;     coth-1  -  =  ^log,  ^— !— 
a  a  —  y  a  y  —  a 

The  anti-gudermannian  of  an  angle  w,  denoted  by  gd""1^,  is  a  number 
defined  by  gd~]w  =  loge  tan  (lAir  +  tfu)  =  J'sec  u  du.  When  u  is  small, 
gd~lu  =  u  +  y*u* 


ANALYTICAL  GEOMETRY 

THE  POINT  AND  THE  STRAIGHT  LINE 

Rectangular  Co-ordinates  (Fig.  1).  Let  Pi  =  (xi,  yi),  P2  =  (22,  1/2). 
Then,  distance  PiP2  =  V(^i  —  ^)2  +  (y\  —  2/2)*;  slope  of  PiP2  =  m  —  tan  u 
=  (Zft  —  2/i)/(£2  —  xi);  co-ordinates  of  mid-point  are  x  =  li(xi  +  xz), 
y  =  Yt(yi  +  2/2);  co-ordinates  of  point  (l/w)th  of  the  way  from  Pi  to  Pa  are 
x  =  xi  +  (1AO(Z2  -  si),  £/  =  2/1  +  (l/n)(2/2  -  3/1). 

Let  mi,  mi  be  the  slopes  of  two  lines;  then,  if  the  lines  are  parallel,  mi  =  m*; 
if  the  lines  are  perpendicular  to  each  other,  mi  =  —  1/mz. 

Equations  of  a  Straight  Line. 

1.  Intercept  Form  (Fig.  2) :  — \-  -  =  1.     (a,  b  =  intercepts  of  the  line  on 

a       o 
the  axes.) 

2.  Slope  Form  (Fig.  3) :     y  =  mx  +  b.     (m  =  tan  u  =  slope;  6  =  inter- 
cept on  the  2/-axis;  see  also  Fig.  2,  p.  174.) 

3.  Normal  Form  (Fig.  4) :     x  cos  v  -f  y  sin  v  =  p.     (p  =  perpendicular 
from  origin  to  line;  v  =  angle  p  makes  with  the  x-axis.) 

4.  Parallel-intercept  Form   (Fig.  5) :    -    — -  =  -  •        (6,  c  =  intercepts  on 

C   ~"~    O  nJ 

two  parallels  at  distance  k  apart.) 


FIG.  1.  FIG.  2.  FIG.  3. 


5.  General  Form:     Ax  +  By  +  C  =  0.     [Here  a  =  -  C/A,  b  =  -  C/B, 
m  =  —  A/B,  cos  v  =  A/R,  sin  v  =  B/R,  p  =  -  C/R,  where  R  =  ±  \/42  +  .Ba 
(sign  to  be  so  chosen  that  p  is  positive).] 

6.  Line  Through  (xi,  yi)  with  Slope  m:     y  —  yi  =  m(x  —  xi). 

7.  Line  Through  (xi,  yi)  and  (xz,  yi):     y  —  yi  =  —  -  -1  (x  —  an). 

Xz  —  zi 

8.  Line  Parallel  to  a;-  axis:     x  =  a;  to  y-axis:    y  =  b. 
Angles  and  Distances. 

If  u  *=  angle  between  two  lines  whose  slopes  are  mi,  mz,  then 
mz  —  mi  If  parallel,  mi  =  mz. 

1  +  mjmi  If  perpendicular,  mim2  =   —  1. 

If  u  =  angle  between  the  lines  Ax  +  By  +  C  =  0  and  A'x  +  J5'i/  +  C'  =  0, 
then 

A  A'  +  BE'  If  parallel,  A  /A'  =  B/B'. 

COS  M    =     —  -  ••'  • 

+  \/(A2  +£2)  (A/2  +5'2)      If  perpendicular,  A  A'  -f  5B'  =  0. 
The  equations  of  the  bisectors  of  the  angles  between  the  two  lines  just 
mentioned  are 


THE  POINT  AND   THE  STRAIGHT  LINE;  THE  CIRCLE 


137 


The  equation  of  a  line  through  (xi,  y\)  and  meeting  a  given  line  y  =  mx  +  b 
at  an  angle  u,  is 

ra  +  tan  u 

y  —  2/i  =  ; (z  —  zO- 

1  —  ra  tan  w 

The  'distance  from  (XQ,  y$)  to  the  line  Ax  +  By  +  C  =  0  is 


where  the  vertical  bars  mean  "the  absolute  value  of." 

The  distance  from  (XQ,  j/o)  to  a  line  which  passes  through  (xi,  y\)  and  makes 
an  angle  u  with  the  z-axis,  is 

•  D  =  (XQ  —  xi)  sin  u  —  (2/0  —  2/0  cos  u. 

Polar  Co-ordinates  (Fig.  6).  Let  (x,  y)  be  the  rec- 
tangular and  (r,  0)  the  polar  co-ordinates  of  a  given 
point  P.  Then  x  =  r  ccys  0;  y  =  r  sin  6;  x*  +  yz  =  r2. 

Transformation  of  Co-ordinates.  If  origin  is  moved 
to  point  (XQ,  2/0),  the  new  axes  being  parallel  to  the  old,  fiQ.  6. 

x  —  XQ  +  x',  y  =  yo  +  y'> 

If  axes  are  turned  through  the  angle  u,  without  change  of  origin, 

x  =  x'  cos  u  —  yf  sin  u,     y  =  x'  sin  u  +  y'  cos  u. 


THE  CIRCLE 

(See  also  pp.  99,  103-105,  106) 

Equation  of  Circle  with  center  (a,6)  and  radiua  r: 
(x  -  a)2  +  (y  -  6)2  =  r2. 

If  center  is  at  the  origin,  the  equation  becomes  x*  +  j/2  =  r2.  If  circle 
goes  through  the  origin  and  center  is  on  the  z-axis  at  point  (r,  0),  equation 
becomes  z2  +  y2  =  2rx.  The  general  equation  of  a  circle  is 

x2  +  2/2  +  Dx  +  Ey  +F  =  0;  it  has  center  at  (  -D/2,  -E/2),  and 

radius  =\/(-D/2)2+  (E/2)2  —  F  (which  may  be  real,  null,  or  imaginary). 

The  equation  of  the  radical  axis  of  two  circles,  x2  +  y2  +  Dx  +  Ey  + 
F  =  0  and  x2  +  y2  +  .D'z  +  JE's/  +  F1  =0,  is  (Z>  -  D')x  +  (E  —  E'}y  + 
(F  —  F')  =  0:  The  tangents  drawn  to  two  circles  from  any  point  of  their 
radical  axis  are  of  equal  length.  If  the  circles  intersect,  the  radical  axis 
passes  through  their  points  of  intersection  (see  p.  100). 

The  equation  of  the  tangent  to  x2  +  y2  =  r2  at  (xi,  2/1)  is  xix  +  y\y  =  r2. 
The   tangent   tp    x2  +  y2  +  Dx  +  Ey  +  F   =  0   at    (xi,   y\)    is 
xix   +  yiy  +  yzD(x  +  a*)  +  ftE(y  +2/1)   +  F  =  0.      The   line   y  =  mx  +  b 
will  be  tangent  to  the  circle  x2  -}-  y*  =*  rz  if  b  =  a\/l  +  m2. 

Equations  of  Circle  in  Parametric  Form.  It  is  sometimes  convenient 
to  express  the  co-ordinates  x  and  y  of  the  moving  point  P 
(Fig.  7)  in  terms  of  an  auxiliary  variable,  called  a  parameter. 
Thus,  if  the  parameter  be  taken  as  the  angle  u  which  the 
radius  OP  makes  with  the  z-axis,  then  the  equations  of  the 
circle  in  parametric  form  will  be  a;  =  acosu;y  =  asinu.  For 
every  value  of  the  parameter  w,  there  corresponds  a  point 
(x,  y)  on  the  circle.  The  ordinary  equation  x2  +  j/2  =  a2  can 
be  obtained  from  the  parametric  equations  by  eliminating  u. 


FIG.  7. 


138 


ANALYTICAL  GEOMETRY 


THE   PARABOLA 

The  parabola  (see  also  p.  107)  is  the  locus  of  a  point  which  moves  so  that  its 
distance  from  a  fixed  line  (called  the  directrix)  is  always  equal  to  its  distance 
from  a  fixed  point  F  (called  the  focus) .  See  Fig.  8.  The  point  half-way  from 
iocua  to  directrix  ia  the  vertex,  O,  The  line  through  the  focus,  perpen- 
dicular to  the  directrix,  is  the  principal  axis.  The  breadth  of  the  curve 
at  the  focus  is  called  the  latus  rectum,  or  parameter,  =  2p,  where  p  is  the 
distance  from  focus  to  directrix.  (Compare  also  Fig.  3,  p.  174..) 


H 

s 

<•—  x  —*\PS 

ip^ 

V 

0 

I 

/YlS 

/      'V   X 

pi      v 

Directrix 

P 
T 

\ 

Fic 

F        M 
p 

N 

u  8. 

A/ 

FIG.  9. 


FIG.  10. 


Any  section  of  a  right  circular  cone  made  by  a  plane  parallel  to  a  tangent  plane 
of  the  cone  will  be  a  parabola. 

Equation  of  Parabola,  origin  at  vertex  (Fig.  8) :     j/2  =  2px. 

Polar  Equation  of  Parabola,  referred  to  F  as  origin  and  Fx  as  axis 
(Fig.  9):  r  =  p/(l  -  cos  6). 

Equation  Referred  to  the  Tangents  at  the  ends  of  the  latus  rectum  a? 
axes  (Fig.  10) :  x^  +  y^  =  o^,  where  a  =  p\/2. 


\ 


FIG.  11. 


FIG.  12. 


FIG.  13. 


The 


Equation  of  Tangent  to  y2  =  2px  at  (xi,yi):  y\y   =  p(x  +  xi). 
line  y  =  mx  +  b  will  be  tangent  to  y*  =  2px  if  6  =  p/(2m). 

The  tangent  PT  at  any  point  P  bisects  the  angle  between  PF  and  PH 
(Fig.  8).  A  ray  of  light  from  F,  reflected  at  P,  will  move  off  parallel  to  the 
principal  axis.  The  subtangent,  TM ,  ia  bisected  at  O.  The  subnormal, 
M  N,  ia  constant,  and  equal  to  p.  The  locus  of  the  foot  of  the  perpendicular 
from  the  focus  on  a  moving  tangent  ia  the  tangent  at  the  vertex  (Fig.  11). 
The  locus  of  the  point  of  intersection  of  perpendicular  tangents  is  the  directrix 
(Fig.  12).  The  Iocua  of  the  mid-points  of  a  set  of  parallel  chords  whose 
slope  is  m  is  a  straight  line  parallel  to  the  principal  axis  at  a  distance 


THE  PARABOLA 


139 


and  is  called  a  diameter  (Fig.  13).  If  M  is  the  mid-point  of  a  chord  PQ, 
and  if  T  is  the  point  of  intersection  of  the  tangents  atP  and  Q,  then  TM  is 
parallel  to  the  principal  axis,  and  is  bisected  by  the  curve  (Fig.  13). 

To  Construct  a  Tangent  to  a  Given  Parabola.  (1)  At  a  given  point  of 
contact,  P  (Fig.  14):  Find  T  so  that  OT  =  OM,  or  FT  =  FP.  Then  TP  is 
the  tangent  at  P.  Or,  make  MAT  =  p  =  2(OF);  ihenPN  is  the  normal  atP. 

(2)  From  a  given  external  point,  Q  (Fig.  15) :  With  Q  as  center  and  radius 
QF  draw  circle  cutting  the  directrix  in  H ;  draw  HP  parallel  to  principal  axis; 
then  P  is  required  point  of  contact.  As  check,  note  that  QP  is  the  perpen- 
dicular bisector  of  FH. 


M      N 


FIG.  14. 


FIG.  15. 


FIG.  16. 


To  Construct  a  Parabola.  1.  GIVEN  ANT  Two  POINTS,  P  AND  Q,  THE 
TANGENT  PT  AT  ONE  OP  THEM,  AND  THE  DIRECTION  OF  THE  PRINCIPAL  Axis 
OX.  In  Fig.  16,  let  K  be  a  variable  point  on  a  line  through  Q  parallel  to 
OX.  Draw  KR  parallel  to  PT  (meeting  PQ  in  R),  and  draw  RS  parallel  to 
OX  (meeting  PK  in  S) ;  then  S  is  a  point  of  the  curve.  NOTE.  A  line  through 
P  parallel  to  the  principal  axis  bisects  all  chords  parallel  to  the  tangent  PT. 

2.  GIVEN  THE  VERTEX  O  AND  Focus  F.  (a)  In  Fig.  17  draw  Oy  perpen- 
dicular to  OF,  and  slide  the  vertex  of  a  right  angle  along  Oy  so  that  one  side 
always  passes  through  F;  then  the  other  side  will  always  be  a  tangent  to  the 
parabola. 


O  ,  F 


FIG.  17. 


FIG.,  18. 


(6)  Take  a  piece  of  paper  (Fig.  18)  with  a  straight  edge,  d,  and  mark  a 
point  F  near  the  edge.  Let  K  be  a  variable  point  of  the  edge,  and  fold  the 
paper  so  that  K  coincides  with  F.  The  crease  will  be  a  tangent  to  the  parabola 
which  has  focus  F  and  directrix  d. 

(c)  In  Fig.  19,  let  M  be  a  variable  point  of  the  principal  axis,  and  lay  off 
MN  =  2(OF)  —  p.  WithF  as  center  and  radius  FN  draw  a  circle,  cutting  the 
perpendicular  at  M  in  P.  Then  P  is  a  point  of  the  curve,  and  PT  and  PN  are 
the  tangent  and  normal  at  P. 

3.  GIVEN  Two  TANGENTS  AND  THEIR  POINTS  OF  CONTACT,  P  AND  Q 
(Fig.  20) .  Divide  TP  and  QT  into  any  number  of  equal  parts  (here  4) .  Then 
the  lines  11,  22,  33,  .  .  .  will  be  tangents  to  the  parabola.  This  method  is 
especially  advantageous  in  drawing  rather  flat  arcs. 


140 


ANALYTICAL  GEOMETRY 


The  Radius  of  Curvature  of  y*  =  2px  at  a  point  P  =  (x,y)  is  R  = 
(p  +  2z)^/\/p,  or  R  =  p/sin3  v,  where  v  =  the  angle  which  the  tangent  at 
P  makes  withPF  (Fig.  21).  At  the  vertex,  R  =  p.  To  construct  the  radius 
of  curvature  at  any  point  P,  lay  off  PR  =  2(PF)  parallel  to  the  principal  axis, 
and  draw  RC  perpendicular  to  the  axis,  meeting  the  normal,  PN,  in  C.  Then 
Cis  the  center  of  curvature  for  the  point  P,  and  a  circle  about  C  with  radius  CP 
will  coincide  closely  with  the  parabola  in  the  neighborhood  of  P. 


FIG.  20. 


FIG.  21. 


THE  ELLIPSE 


The  ellipse  (see  also  p.  107)  has  two  foci,  F  and  F'  (Fig.  22) ,  and  two  direc- 
trices, DH  and  D'H'.  If  P  is  any  point  of  the  curve.PF  +PF'  is  constant,  =2a; 
and  PF/PH  (or  PF'/PH')  is  also  constant,  -e,  where  e  is  the  eccentricity 
(e<l).  Either  of  these  properties  may  be  taken  as  the  definition  of  the 
curve.  The  relations  between  e  and  the  semi-axes  a  and  b  are  as  shown  in 
Fig.  23.  Thus,  62  =  o2(l  -  e2),  ae  =  Va2  -  V,  e*  =  1  -  (6/a)2.  The 
semi-latus  rectum  =  p  =  a(l  —  e2)  =  62/o.  Note  that  b  is  always  less 
than  a,  except  in  the  special  case  of  the  circle,  in  which  6  =  a  and  e  =  0. 

y 


FIG.  23. 


FIG.  24. 


FIG.  25. 


Any  section  of  a  right  circular  cone  made  by  a  plane  which  cuts  all  the 
elements  of  one  nappe  of  the  cone  will  be  an  ellipse;  if  the  plane  is  perpen- 
dicular to  the  axis  of  the  cone,  the  ellipse  becomes  a  circle. 

Equation  of  Ellipse,  center  as  origin: 

—  +_-=!,   or  y  —  ±  — \/o2  —  a;2. 
a         o  a 

If  P  =  (x,  y)  ia  any  point  of  the  curve,  PF  -  a  +  ex,  PF'  =  a  "-  ex. 

Equations  of  the  Ellipse  in  Parametric  Form:  x  =  a  cos  u,  y  = 
6  sin  u,  where  u  is  the  eccentric  angle  of  the  point  P  =  (z.j/).  See  Fig.  28. 


THE  ELLIPSE 


141 


Polar  Equation,  focus  as  origin,  axes  as  in  Fig.  24:    r  =  p/(l  —  e  cos  0). 

Equation  of  the  Tangent  at  (xi,yi) :  b*xix  +  a*yiy  «=  a262. 

The  line  y  =  mx  +  k  will  be  a  tangent  if  k  =  +  A/o2m2  +  &2.  The  normal 
at  any  point  P  bisects  the  angle  between  PF  and  PF"  (Fig.  25).  The  locus  of 
the  foot  of  the  perpendicular  from  the  focus  on  a  moving  tangent  is  the  circle 
on  the  major  axis  as  diameter  (Fig.  26).  The  locus  of  the  point  of  intersection 
of  perpendicular  tangents  is  a  circle  with  radius  V  a2  +  62  (Fig.  27). 


FIG.  26. 


FIG.  27. 


FIG.  28. 


FIG.  29. 


Ellipse  as  a  Flattened  Circle.  Eccentric  Angle.  If  the  ordinates  in 
a  circle  are  diminished  in  a  constant  ratio,  the  resulting  points  will  lie  on 
an  ellipse  (Fig.  28).  If  Q  traces  the  circle  with  uniform  velocity,  the  corre- 
sponding point  P  will  trace  the  ellipse,  with  varying  velocity.  The  angle  u  in 
the  figure  is  called  the  eccentric  angle  of  the  point  P. 

Conjugate  Diameters  are  lines  through  the  center,  each  of  which  bisects 
all  the  chords  parallel  to  the  other  (Fig.  29).  If  mi  and  mz  are  the  slopes,  then 
m\mz  =  —  &2/o2.  One  pair  of  conjugate  diameters  are  the  diagonals  of  the 
rectangle  circumscribing  the  ellipse.  The  eccentric  angles  of  the  ends  of  two 
conjugate  diameters  differ  by  90  deg.  Thus  (Fig.  30),  if  CQ  and  CQ'  are 
perpendicular  radii  in  the  circle,  CP  and  CPf  will  be  conjugate  semi-diameters 
in  the  ellipse.  A  parallelogram  formed  by  tangents  drawn  parallel  to  a  pair  of 
conjugate  diameters  has  a  constant  area,  =  4a&  (Fig.  31).  Also,  if  a', b' 
are  conjugate  semi-diameters,  and  w  the  angle  between  them,  then  a'2  -f  &'2  = 
a2  +  62  and  a'b'  =  a&/sin  w. 


FIG. 


FIG.  31. 


FIG.  32. 


FIG.  33. 


To  Construct  a  Tangent  to  a  Given  Ellipse.  (1)  AT  A  GIVEN  POINT  OF 
CONTACT,  P.  Bisect  the  angle  between  the  focal  radii  PF  and  PF'  (Fig.  25). 

(2)  FROM  A  GIVEN  EXTERNAL  POINT,  R.  (a)  Through  R  draw  any  two 
lines  cutting  the  ellipse,  one  in  A  and  B,  the  .other  in  C  and  D  (Fig.  32). 
Through  the  point  of  intersection  of  AD  and  BC  and  the  point  of  intersec- 
tion of  AC  and  BD,  draw  a  line  cutting  the  ellipse  in  P  and  Q.  Then  P  and  Q 
are  the  required  points  of  contact.  (&)  With  R  as  a  center  and  radius  RF, 
draw  an  arc;  with  F'  as  center  and  radius  2a  draw  an  arc,  intersecting  the 
first  in  S;  and  let  SF'  meet  the  curve  in  T.  Then  T  is  the  point  of  contact 
(Fig.  33). 


142 


ANALYTICAL  GEOMETRY 


To  Construct  an  Ellipse,  Given  a  and  b.  (1)  In  Fig.  34,  with  O  as 
center,  draw  circles  with  radii  a  and  b  (and  also  a  third  circle  with  radius 
0+6).  Let  a  variable  ray  through  O  cut  these  circles  in  J,  K  (and  5) ; 
through  J  and  K  draw  parallels  to  the  axes,  meeting  in  P.  Then  P  is  a  point 
of  the  ellipse  (and  SP  is  the  normal  atP). 

(2)  In  Fig.  35,  let  P  divide  a  line  AB  so  that  PA  =  a  and  P5  =  6.  Then  if 
A  and  JS  slide  on  the  axes,  P  will  describe  an  ellipse. 


FIG.  34. 


8 

FIG.  35. 


FIG.  36. 


(3)  In  Fig.  36,  let  PBA  be  a  straight  line  such  that  PA  =  a  and  PB  =  b. 
Then  if  A  and  B  slide  on  the  axes,  P  will  trace  an  ellipse.     (Use  a  strip  of 
paper,  with  the  points  P,  B,  and  A  marked  on  it.) 

(4)  Find  the  foci,  F  and  F',  by  striking  an  arc  of  radius  a  with  center  at  B 
(Fig.  37) .     Drive  pins  atF,  F',  and  B,  and  adjust  a  loop  of  thread  around  them. 
Then  remove  the  pin  at  B,  and  replace  it  by  a  pencil  point;  by  moving  the 
pencil  so  as  to  keep  the  string  taut,  the  complete  ellipse  can  be  drawn  at  one 
sweep.     Or,  use  a  mechanical  ellipsograph. 

(5)  and  (6).     Apply  methods  (1)  and  (2)  of  the  following  paragraph  to 
the  special  case  in  which  OP  and  OQ  are  perpendicular  semi-axes. 


FIG.  37. 


FIG.  38. 


FIG.  39. 


To  Construct  an  Ellipse,  Given  a  Pair  of  Conjugate  Semi-diameters, 
OP  and  OQ.  (1)  Complete  the  parallelogram,  as  in  Fig.  38.  Divide  QD 
and  QO  into  n  equal  par ts,  1,  2,3,  .  .  .  and  1',  2',  3',  .  .  .  ConnectPwith 
1,  2,  3,  .  .  .  andP'  with  1',  2',  3'  .  ...  The  points  of  intersection  of  corrfe- 
sponding  lines  will  be  points  of  the  ellipse. 

(2)  Take  any  point  K  onPQ  (Fig.  39).     Draw  EKU,  and  draw  KV  parallel 
to  OP.     Then  UV  will  be  a  tangent.    By  varying  K  alongPQ  as  many  tangents 
may  be  drawn  as  desired,  thus  "enveloping"  the  ellipse. 

(3)  Through  P  (Fig.  40),  draw  a  perpendicular  PT  to  OQ,  and  lay  off  PR  = 
PS  =  OQ.     Then  if  the  line  RPT  is  made  to  slide  with  one  end  on  OR  and  the 
other  on  OQ,  P  will  trace  the  ellipse.     Further,  the  bisectors  of  the  angle 
ROS  show  the    directions  of    the    principal   axes,    and   OR  +  OS  =  2a  and 


THE  ELLIPSE 


143 


OR  —  OS  =*  26.  Also,  if  a  line  through  P  perpendicular  to  RS  (and  there- 
fore tangent  to  the  ellipse  atP)  meets  the  minor  axis  in  M,  a  circle  with  M  as 
center  and  MR  or  MS  as  radius  will  cut  the  major  axis  in  the  two  foci. 


FIG.  40. 


FIG.  41. 


FIG.  42. 


To  Construct  an  Ellipse  Approximately  by  Circular  Arcs.  [Methods 
(1)  and  (2)  employ  two  radii,  (3)  and  (4)  employ  three  radii.]  (1)  In  Fig.  41, 
lay  off  OL  =  OA  and  BS  =  BL  =  a  -  b.  Bisect  SA  in  T,  and  draw  THK 
perpendicular  to  BA.  Then  H  is  one  center,  with  radius  HA,  and  K  is  the 
other  center,  with  radius  KB.  The  junction  point  Q  of  the  two  arcs  will  fall  a 
little  outside  the  true  ellipse. 

(2)  In  Fig.  42,  lay  off  OU  =  0V  =  OB  =  b.  Draw  UG  perpendicular  to 
the  axis  and  DG  at  45°.  With  G  as  center  draw  an  auxiliary  arc  with  radius 


FIG.  43. 


FIG.  44. 


=»  AV  =  a  —  b,  and  through  D  draw  DMN  just  touching  this  arc.  Then  M 
is  one  center  (with  radius  M  A)  and  N  is  the  other  center  (with  radius  NB) . 
The  junction  point  P  of  the  two  arcs  will  be  a  true  point  of  the  ellipse.  [E.  V. 
Huntington.] 

(3)  Through  D  (Fig.  43)  draw  DCiC3  perpendicular  to  AB.  Call  CiA  =  n 
and  CzB  =  r3.  Lay  off  BE  =  BO  (  =6),  and  on  ED  as  diameter  draw  a  semi- 
circle cutting  the  minor  axis  in  W',  then  BW  =  \/o6  =  r2.  Lay  off  AZ  = 


144 


ANALYTICAL  GEOMETRY 


BW.  From  Ci  with  radius  C\Z(=  rz  —  n),  and  from  £3  with  radius  C&W 
(  =r3  —  ra),  draw  arcs  intersecting  in  Cz.  Draw  CzCz  extended  and  CzCi  ex- 
tended. Then  draw  in  the  three  arcs,  with  centers  at  Ci,  €2,  €3  and  radii  n, 
rz,  ra.  NOTES.  Since  n  and  r$  are  the  radii  of  curvature  of  the  ellipse  at  A 
and  B,  this  construction  gives  a  curve  which  is  a  little  too  sharp  at  A  and 
a  little  too  flat  at  B.  A  more  accurate  construction  is  the  folio  wing: 

(4)  In  Fig.  44,  lay  off  BE  =  BH  —  BO  =  b.  Through  the  mid-point  X 
of  BE  draw  XG  perpendicular  to  the  axis,  and  through  D  draw  DG  at  an  angle 
of  45  deg.  From  G  as  center  draw  an  auxiliary  arc  with  radius  =  DH 
(=  a  —  b),  and  through  D  draw  DCiCz  just  touching  this  arc.  Take  CiA 
as  n  and  CzB  as  ra.  On  DE  as  diameter  draw  a  semi-circle  cutting  the  minor 
axis  in  W,  and  take  BW(=\/ab)  as  r2.  Lay  off  AZ  =  BW.  From  Ci 
with  radius  CiZ(  =  rz  —  n),  and  from 
Cs  with  radius  CzW(  =  r%  —  rz),  draw 
arcs  intersecting  in  Cz.  Then  Ci, 
Cz,  Ca  are  the  required  centers.  [E. 
V.  Huntington.] 

Radius  of  Curvature  of  Ellipse 
at  Any  Point  P  =  (x,  y)  is  R  = 
o262(o;2/a4  +  yz/b*)M  =  p/sin3v,  where 
v  is  the  angle  which  the  tangent  at  P 
makes  with  PF  or  PF'.  At  end  of 
major  axis,  R  =  62/a  =  MA;  at  end 
of  minor  axis,  R  =  a?/b  =  NB  (see 
Fig.  45).  To  construct  the  radius  JTIG>  45^  JTIG  45. 

of  curvature  at  any  other  point  P 

(Fig.  46) ,  draw  the  normal  at  P  (by  bisecting  the  angle  between  PF  and  PF') 
and  let  it  meet  the  major  axis  in  N.  At  N  draw  a  perpendicular  toPN  meet- 
ing PF  in  H .  At  H  draw  a  perpendicular  to  PH  meeting  PN  in  C.  Then  C 
is  the  center  of  curvature  for  the  point  P,  and  a  circle  about  C  with  radius 
CP  will  coincide  closely  with  the  ellipse  in  the  neighborhood  of  P.  [Note. 
If  the  circle  of  curvature  meets  the  ellipse  in  Q,  then  the  tangent  at  P  and 
the  line  PQ  are  equally  inclined  to  the  axis.] 

THE  HYPERBOLA 

The  hyperbola  (see  also  p.  107)  has  two  foci,  F  and  F',  at  distances  ±  ae 
from  the  center,  and  two  directrices,  DH  and  D'H',  at  distances  ±  a/e  from 


FIG.  47. 


FIG 


the  center  (Fig.  47).  If  P  is  any  point  of  the  curve,  \PF  —  PF'\  is  constant, 
=  2a;  &ndPF/PH  (orPF'/PH')  is  also  constant,  =  e  (called  the  eccentricity), 
where  e  >  1.  Either  of  these  properties  may  betaken  as  the  definition  of  the 


THE  HYPERBOLA 


145 


curve.  The  curve  has  two  branches  which  approach  more  and  more  nearly 
two  straight  lines  called  the  asymptotes.  Each  asymptote  makes  with  the 
principal  axis  an  angle  whose  tangent  is  b/o.  The  relations  between  e,  a,  and 
b  are  shown  in  Fig.  48:  b2  =  o2(e2  —  1),  ae  =  Va2  +  b2,  e2  =  1  +  (b/a)2. 
The  semi-latus  rectum,  or  ordinate  at  the  focus,  is  p  =  o(e2  —  1)  =b2/a. 

Any  section  of  a  right  circular  cone  made  by  a  plane  which  cuts  both  nappes 
of  the  cone  will  be  a  hyperbola.  (Compare  also  Fig.  3,  p.  174.) 

Equation  of  the  Hyperbola,  center  as  origin: 

—«  -  r,  =  1.  or  y  •• 


a*       b* 

If  P  =  (x,y)  is  on  the  right-hand  branch,  PF  =  ex— a,  PF'  =  ex+a. 
If  P  is  on  the  left-hand  branch,  PF  =  -ex  +a,  PF'  =  -ex  -a. 

Equations  of  Hyperbola  in  Parametric  Form.  (1)  x  =  a  cosh  u, 
y  —  b  sinh  u.  (For  tables  of  hyperbolic  functions,  see  pp.  60  and  61.)  Here 
u  may  be  interpreted  as  A  /a2,  where  A  is  the  area  shaded  in  Fig.  49. 


FIG.  49. 


FIG.  50. 


(2)   x  =  a  sec  v,  y  =  b  tan  v,  where  v  is  an  auxiliary  angle  of  no  special 
geometric  interest. 

Polar  Equation,  referred  to  focus  as  origin,  axes  as  in  Fig.  50: 

r  =  p/(l  —  e  cos  6). 

Equation  of  the  Tangent  at  (zi,2/i):  b*xix  —a*yiy  =o2b2. 
The  line  y  =  mx  +  k  will  be  a  tangent  if  k  =  ±  VVm2  —  62.     Thetan- 


FIG.  51. 


FIG.  52. 


FIG.  53.  \ 


gent  at  any  point  P  (Fig.  51)  bisects  the  angle  between  PF  andPF'.  The  locus 
of  the  foot  of  the  perpendicular  from  the  focus  on  a  moving  tangent  is  the 
circle  on  the  principal  axis  as  diameter  (Fig.  52).  The  locus  of  the  point  of 
intersection  of  perpendicular  tangents  is  a  circle  with  radius  \/fl2  —  b2,  which 
will  be  imaginary  if  6  >  a  (Fig.  53). 
10 


146 


ANALYTICAL  GEOMETRY 


Properties  of  the  Asymptotes.  (Fig.  54. )  If  P  is  any  point  of  the  curve, 
the  product  of  the  perpendicular  distances  from  P  to  the  two  asymptotes  is  con- 
stant, =  o262/(a2  +  62).  Also,  the  product  of  the  oblique  distances  (the  dis- 
tance to  each  asymptote  being  measured  parallel  to  the  other)  is  constant,  and 
equal  to  }4(o2  +  bz).  If  a  line  cuts  the  hyperbola  and  its  asymptotes,  the 
parts  of  the  line  intercepted  between  the  curve  and  the  asymptotes  are  equal. 
The  part  of  a  tangent  intercepted  between  the  asymptotes  is  bisected  by  the 
point  of  contact.  The  triangle  bounded  by  the  asymptotes  and  a  variable 
tangent  is  of  constant  area,  =  ab.  If  a  line  through  Q  perpendicular  to  the 
principal  axis  meets  the  asymptotes  in  R  and  S  (see  Fig.  54),  then  QR  X  QS  — 
b*.  If  a  line  through  Q  parallel  to  the  principal  axis  meets  the  asymptotes  in 
U  and  V,  then  QU  X  QV  =  a2. 


FIG.  54. 


FIG.  55. 


Conjugate  Hyperbolas  are  two  hyperbolas  having  the  same  asymptotes 
with  semi-axes  interchanged  (Fig.  55).    The  equation  of  the  hyperbola  conju- 

x*       y*  x*       y2 

gate  to -2  --2  =  1,  is  -2  -- 


1. 


Conjugate  Diameters  are  lines  through  the  center,  each  of  which  bisects 
all  the  chords  parallel  to  the  other — a  chord  which  does  not  meet  the  given 
hyperbola  being  understood  to  be  terminated  by  the  conjugate  hyperbola 
(Fig.  65).  If  mi  and  m*  are  the  slopes,  then  rmmz  =  62/a2.  Each  asymptote, 
regarded  as  a  diameter,  is  its  own  conjugate.  If  a  parallelogram  is  formed 
by  tangents  drawn  parallel  to  a  pair  of  conjugate  diameters,  its  vertices  will 
lie  on  the  asymptotes,  and  its  area  will  be  constant  =  4a&.  If  a',  b'  are 
conjugate  semi-diameters,  and  w  the  angle  between  them,  then  a/2  —  b'2 
=  a2  —  62,  and  a'b'  =  ab/sin  w. 

Equilateral  Hyperbola  (a  =6).  Equation  referred  to  principal  axes 
(Fig.  56) :  x*  —  y*  =  a2.  NOTE,  p  =  a.  Equation  referred  to  asymptotes 
as  axes  (Fig.  57) :  xy  =  o2/2.  (See  also  Fig.  3,  p.  174.) 

Asymptotes  are  perpendicular.  Eccentricity  =  \/2.  Any  diameter  is  equal 
in  length  to  its  conjugate  diameter. 

y 


Fio.  66. 


FIG.  67. 


THE  CATENARY 


147 


To  Construct  a  Tangent  at  any  given  point  P  of  a  hyperbola.  In  Fig.  58, 
draw  PA  and  PS  parallel  to  the  asymptotes,  and  take  OS  =»  ?(OA)  and  OT  = 
2(OB).  Then  ST  is  the  tangent  at  P. 


FIG.  58. 

To  Construct  a  Hyperbola^   given  the  asymptotes  and  any  point  P. 

(1)  In  Fig.  59  let  TPT'  bea  variable  line  throughP,  andlayoff  T'Pf  =  TP; 
then  P'  is  a  point  of  the  curve. 

(2)  In  Fig.  60,  draw  PA  and  PB  parallel   to  the  asymptotes.     Lay  off 
OA'  =  n(OA)  and  OB'  =  (\/ri)(OB},  where  n  is  any  number;  and  throughA' 
and  Bf  draw  parallels  to  the  axes;  these  will  meet  in  a  point  P'  of  the  curve. 


FIG.  59. 


FIG.  61. 


(3)  (Fig.  61.)  Take  any  point  K  in  the  ordinate  PM,  and  draw  OK 
meeting  the  line  through  P  parallel  to  the  z-axis  in  R.  Draw  a  parallel  to 
the  x-axis  through  K  and  a  parallel  to  the  y-axia  through  R,  meeting  in  Q. 
Then  Q  is  a  point  of  the  curve. 

THE  CATENARY 

The  catenary  is  the  curvein  which  aflexible  chain  or  cord  of  uniform  density 
will  hang  when  supported  by  the  two  ends.  Let  w  = 
weight  of  the  chain  per  unit  length ;  T  =  the  tension 
at  any  point  P;  and  Th,Tv  =  the  horizontal  and 
vertical  components  of  T.  The  horizontal  com- 
ponent Th  is  the  same  at  all  points  of  the  curve. 

The  length  a  =  Th/w  is  called  the  parameter  of  the 
catenary,  or  the  distance  from  the  lowest  point  O  to 
the  directrix  DQ  (Fig.  62).  When  a  is  very  large, 
the  curve  is  very  flat.  For  methods  of  finding  o  in 
any  given  case,  see  problems  1-6  below. 

The  rectangular  equation,  referred  to  the  lowest 
point  as  origin,  is  y  —  a  [cosh  (x/a)  —  1].  (For 
table  of  hyperbolic  functions,  see  p.  60.)  In  case  of 


FIG.  62. 


148 


ANALYTICAL  GEOMETRY 


x2 
very  flat  arcs  (a  large),  y  =  - 


.  .  .  ;  s  =  x  +  ^  —  + 


approximately, 


so  that  in  such  a  case  the  catenary  closely  resembles  a  parabola. 

If  the  perpendicular  from  O  to  the  tangent  at  P  meets  the  directrix  in  Q, 
then  DQ  =  arc  OP  =  s  andOQ  =  y  +  o.  The  radius  of  curvature  atP  is 
R  =  (y  +o)2/o,  which  is  equal  in  length  to  the  portion  of  the  normal  inter- 
cepted between  P  and  the  directrix. 

Problems  on  the  Catenary  (Fig.  62).  When  any  two  of  the  four 
quantities  x,  y,  s,  T/w  are  known,  the  remaining  two,  and  also  the  para- 
meter a,  can  be  found,  as  follows: 

(  1)  GIVEN  x  AND  y.  Compute  y/x,  and  find  from  Table  1  the  value  of  the 
auxiliary  variable  z.  Then  compute  a  =  x/z,  s  =  a  sinh  z,  and  T  — 
wa  cosh  z.  Or,  having  z,  find  s/x  and  wx/T  by  using  Tables  3  and  2  inversely, 
and  hence  (since  x  is  known)  compute  s  and  T/w  without  the  use  of  o. 


TABLE  1.     GIVING  z  WHEN  y/x  is  KNOWN.     THEN  a  =  x/z 


y/x 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

00 

0.0000 

0.0200 

0.0400 

0.0600 

0.0800 

0.0999 

n 

.1199 

0.1398 

0.1597 

0.1795 

01 

0.1993 

0.2191 

0.2389 

0.2586 

0.2782 

0.2978 

0 

.3173 

0.3368 

0.3562 

0.3756 

02 

0.3948 

0.4140 

0.4332 

0.4522 

0.4712 

0.4901 

r 

.5089 

0.5276 

0.5463 

0.5648 

0.3 

0.5833 

0.6016 

0.6199 

0.6381 

0.6561 

0.6741 

n 

.6919 

0.7097 

0.7274 

0.7449 

0  4 

0.7623 

0.7797 

0.7969 

0.8140 

0.8311 

0.8480 

n 

.8647 

0.8814 

0.8980 

0.9145 

05 

0.9308 

0.9471 

0.9632 

0.9792 

0.9951 

1.0109 

i 

.0266 

1.0422 

1  .0576 

1.0730 

0.6 

1.0883 

1.1034 

1.1184 

1.1334 

1.1482 

1.1629 

i 

.1775 

1.1920 

1.2064 

1.2207 

NOTE. 

y/x  - 

(cosh  z  — 

DA- 

(2)  GIVEN  x  AND  T/w.  Compute  wx/T,  and  find  from  Table  2  the  value 
of  the  auxiliary  variable  z.  Then  compute  a  =  x/z,  y  =  a. (cosh  z  —  l)and 
a  =  a  sinh  z.  Or,  having  z,  find  y/x  and  s/x  by  using  Tables  1  and  3  inversely, 
and  hence  (since  x  is  known)  compute  y  and  s  without  the  use  of  a. 

TABLE  2.     GIVING  z  WHEN  wx/T  is  KNOWN.     THEN  a  =  x/z 


wx/T 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

0.0 
0.1 
0.2 
0.3 
0.4 
0.5 
0.6 

0.0000 
0.1005 
0.2042 
0.3150 
0.4392 
0.5894 
0.8053 

0.0100 
0.1107 
0.2149 
0.3267 
0.4528 
0.6068 
0.8357 

0.0200 
0.1209 
0.2256 
0.3385 
0.4666 
0.6249 
0.8695 

0.0300 
0.1311 
0.2365 
0.3505 
0.4806 
0.6436 
0.9082 

0 
0 
0 
0 
0 
0 
0 

0400 

1414 
2474 
3626 
4950 
6632 
9541 

0.0501 
0.1517 
0.2584 
0.3749 
0.5097 
0.6836 
1.0132 

0.0601 
0.1621 
0.2695 
0.3874 
0.5248 
0.7051 
1.1110 

0.0702 
0.1725 
0.2807 
0.4000 
0.5403 
0.7277 

0.0803 
0.1830 
0.2920 
0.4129 
0.5562 
0.7517 

00904 
0.1936 
0.3035 
0.4259 
0.5726 
0.7775 

NOTE.     wx/T  =  z/cosh  z.     For  every  value  of  wx/T  there  are  two  values  of  z,  one 
less  than  1.200  and  one  greater  than  1.200.     Only  the  smaller  of  these  values  is  tabulated. 


(3)  GIVEN  x  AND  s.  Compute  s/x,  and  find  from  Table  3  the  value  of 
the  auxiliary  variable  z.  Then  compute  a  =  x/z,  y  —  a  (cosh  z  —  l),and 
T  =  wa  cosh  z.  Or,  having  z,  find  y/x  and  wx/T  by  using  Tables  1  and  2 
inversely,  and  hence  (since  x  is  known)  compute  y  and  T/w  without  the  use 
of  a. 


THE  CATENARY 


149 


TABLE  3.     GIVING  z  WHEN  s/x  is  KNOWN.     THEN  a  =  x/z 


•/* 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

1.000 

0  0245 

0.0346 

0  0424 

0.0490 

0.0548 

0.0600 

0.0648 

0.0693 

0.0735 

1 

6  !  0774 

0.0812 

0.0848 

0.0883 

0.0916 

0 

0948 

0.0980 

0.1010 

0.1039 

0.1067 

2 

0.1095 

0.1122 

0.1149 

0.1174 

0.1200 

0 

1224 

0.1249 

0.1272 

0.1296 

0.1319 

3 

0.1341 

0.1363 

0.1385 

0.1407 

0.1428 

0 

1448 

0.1469 

0.1489 

0.1509 

0.1529 

4 

0.1548 

0.1567 

0.1586 

0.1605 

0.1623 

0 

1642 

0.1660 

0.1678 

0.1696 

0.1713 

1.005 

0.1731 

0.1748 

0.1765 

0.1782 

0.1799 

o 

1815 

0.1831 

0.1848 

0.1864 

C.1880 

6 

0.1896 

0.1911 

0.1927 

0.1942 

0.1958 

0 

1973 

0.1988 

0.2003 

0.2018 

0.2033 

7 

0.2047 

0.2062 

0.2076 

0.2091 

0.2105 

0 

2119 

0.2133 

0.2147 

0.2161 

0.2175 

8 

0.2188 

0.2202 

0.2215 

0.2229 

0.2242 

0 

2255 

0.2269 

0.2282 

0.2295 

0  2308 

9 

0.2321 

0.2334 

0.2346 

0.2359 

0.2372 

0 

2384 

0.2397 

0.2409 

0.2421 

0.2434 

1.01 

0.2446 

0.2565 

0.2678 

0.2787 

0.2892 

0.2993 

0.3091 

0.3186 

0.3278 

3367 

2 

0.3454 

0.3539 

0.3621 

0.3702 

0.3781 

0 

3859 

0.3934 

0.4009 

0.4082 

U.4153 

3 

0.4224 

0.4293 

0.4361 

0.4428 

0.4494 

0 

4559 

0.4623 

0.4686 

0.4748 

0.4809 

4- 

0.4870 

0.4930 

0.4989 

0.5047 

0.5105 

0 

5162 

0.5218 

0.5274 

0.5329 

0.5383 

1.05 

0.5437 

0.5490 

0.5543 

0.5595 

0.5647 

0 

5698 

0.5749 

0.5799 

0.5849 

0.5898 

6 

0.5947 

0.59% 

0.6044 

0.6091 

0.6139 

0 

6186 

0.6232 

0.6278 

0.6324 

0.6369 

7 

0.6414 

0.6459 

0.6504 

0.6548 

0.6591 

0 

6635 

0.6678 

0.6721 

0.6763 

0.6806 

8 

0.6848 

0.6889 

0.6931 

0.6972 

0.7013 

0 

7053 

0.7094 

0.7134 

0.7174 

0.7213 

9 

0.7253 

0.7292 

0.7331 

0.7369 

0.7408 

0 

7446 

0.7484 

0.7522 

0.7559 

0.7597 

1.10 

0.7634 

NOTE: 

s/x  = 

sinh  z/z 

(4)  GIVEN  y  AND  a.  Then 


•S  +  l  *•:(?-») 


tanh-1    | -I  i 


(5)  GIVEN  y  AND  T/w.  Then  a  = y,  x 

w 


y  I  cosh' 


T/w 
(T/w)  -  y 


s  =  V2y(T/w)   -  y*.     Or,  if  y/(T/w)  is  small, 

7    wi 
'  12  ~T 


„  —  x       \  wy 
,     —- —  =  g  -y  ,  approximately, 


(6)   GIVEN  s  AND  T/w.      Then  a;  =-  A  1  -  (  — 


Given  the  Length  2L  of  a  Chain  Supported  at  Two  Points  A  and 
B  not  in  the  Same  Level,  to  find  a.  (See  Fig.  63;  b  and  c  are  supposed 
known.)  Let  ( \/I/2  —  b2) /c  =  s/x;  enter  Table  3  with  this  value  of  s/x,  and 
find  the  corresponding  value  of  the  auxiliary  variable  z.  Then  a  =  c/z. 


150 


ANALYTICAL  GEOMETRY 


NOTE.  The  co-ordinates  of  the  mid-point  M  of  AB  (see  Fig.  63)  are  XQ  = 
a  tanh"1  (b/L),  y0  =  (L/tanh  z)  —  a,  so  tha't  the  position  of  the  lowest  point 
is  determined. 

Correction  for  Sag  in  Chaining  TTphill  (Fig.  64).  Let  I  =  length  of 
tape  (corrected  for  stretch  and  temperature),  w  =  weight  per  unit  length  of 
tape,  A  =  angle  between  the  chord  AB  and  the  horizontal. 


FIG.  63. 


FIG.  64. 


If  the  tension  P  at  the  upper  end  is  known,  compute  wl/P  and  find  k  from 
Table  4.  If  the  tension  Q  at  the  lower  end  is  known,  compute  wl/Q  and  find 
k  from  Table  5.  In  either  case,  chord  AB  =  l  —  kl. 


TABLE  4.     GIVING  k 


TABLE  5.     GIVING  k 


wl 


A=0°  10°  20°  30°  40°  50°  60°  70° 


A  =0°  10°  20°  30°  40°  50°  60°  70° 


.00000  000  000  000  000  000  000  000  000 

002  002  001  001   001   001   000  000  000 

004  004  003  003  002  002  001   000  000 

007  006  006  005  004  003  002  001   000 

Oil  010  009  008  006  004  003  001   000 

.00015  015  013  012  009  006  004  002  000 

020  020  018  016  012  009  005  003  001 

027  026  024  021  016  012  007  003  001 

034  033  031  026  021   015  009  004  001 

042  041  038  033  026  019  Oil   005  001 

.00051  050  046  040  032  023  014  007  002 

060  060  055  048  038  027  017  008  002 

070  070  065  057  045  032  020  009  002 

082  081   076  066  053  038  023  Oil   003 

094  094  087  076  061   044  027  013  003 

00107  107  100  087  070  050  031   015  004 

'     121  121    113  099  079  057  035  017  004 

136  136  128  112  090  065  040  019  005 

151  152  143  125  101   073  045  021  006 

168  168  159  140  113  082  050  024  006 


.00000  000  000  000  000  000  000  000  000 
002  002  001  001  001  001  000  000  000 
004  004  003  003  002  001  001  000  000 
007  006  006  005  004  003  002  001  000 
Oil  010  009  008  006  004  002  001  000 

.00015  014  013  Oil  008  006  004  002  000 

020  020  018  015  Oil  008  005  002  001 

027  026  023  019  015  Oil   006  003  001 

034  032  029  024  019  013  008  004  001 

042  040  036  030  023  016  010  004  001 

.00051  048  043  036  028  019  Oil  005  Oof 
060  057  051  043  033  023  014  006  002 
070  067  060  050  038  026  016  007  002 
082  078  069  057  044  030  018  008  002 
094  089  079  066  050  035  021  010  002 

.00107  101  090  074  057  039  022  01 1  003 

121  114  101  084  064  044  026  012  003 

136  128  113  092  071  049  029  013  003 

151  142  125  103  079  054  032  015  004 

168  157  138  114  087  060  035  016  004 


NOTE,     k  =•  1  -   {[l-\/l  -  2m    sin  u  + 
given  by 


sin  A]},  where  m  =  wl/P  and  u  is 


[1  —  \/l  —  2m  sin  u  +  m2]  secw  =  [sinh  -1  (tanw)  —  sinh  -1  (t&nu  —  m  sec  w)]tan  A. 

Also,  Q  =  P  —  wl  (1  —  k)  sin  A,  where  k  is  the  value  in  Table  4  corresponding  to 
the  given  values  of  P  and  A. 

Correction  for  Stretch  m  Chaining  Uphill.  Let  L  =  unstretched  length 
of  tape  at  working  temperature,  w  =  weight  per  unit  length  of  tape,  A  =  angle 


OTHER   USEFUL  CURVES 


151 


between  chord  AB  and  the  horizontal,  F  =  area  of  cross-section,  E  =  Young's 
modulus  of  elasticity  (for  steel,  E  =  29,000,OOQ  Ib.  per  sq.  in.),  I  =  stretched 
length  (along  curve). 

If  the  tension  P  at  the  upper  end  is  known,  compute  wL/P  and  find  m  from 
Table  6.  Then  I  =  L  +  (LP/FE)  (1  •*•  m). 

If  the  tension  Q  at  the  lower  end  is  known,  compute  wL/Q  and  find  nfrom 
Table  7.  Then  I  =  L  +  (LQ/FE)  (1  +  n) . 

TABLE  6.'  GIVING  m  TABLE  7.     GIVING  n 


«'L 
?- 

4oT  10°  20°  30°  40°  50°  60°  70°  80°  90° 

wL 

~Q 

A=  10°  20°  30°  40°  50°  60°  70°  80°  90° 

.00 
.10 
.20 

.000  .000  .000  .000  .000  .000  .000  .000  .000  .000 
.001  .010.018.026.033.039.044.047.049.050 
.003  .021  .038  .053  .067  .078  .088  .094  .099.  100 

.00 
.10 
.20 

.000.000.000.000.000.000.000.000  000 
.008.016.024.032.038.043.047.049.050 
.014  .03  1  .047  .062  .075  .086  .094  .099  .  1  00 

OTHER  USEFUL  CURVES 

The  Cycloid  is  traced  by  a  point  on  the  circumference  of  a  circle  which  rolls 
without  slipping  along  a  straight  line.  Equations  of  cycloid,  in  parametric 
form  (axes  as  in  Fig.  65)  \}x  =  a  (rad  u  —  sin  w),  y  =  a(l  —  cos  w),  where  a  is 


FIG.  65. 


FIG.  66. 


the  radius  of  the  rolling  circle,  and  rad  u  is  the  radian  measure  of  the  angle  u 
through  which  it  has  rolled.  The  tangent  and  normal  at  any  point  pass 
through  the  highest  and  lowest  points  of  the  corresponding  position  of  the 
generating  circle.  The  radius  of  curvature  at  any  point  P  is  PC  = 
4a  sin(w/2)  .=  2\/2ay  =  twice  thelength  of  the  normal,  PJV.  Theevolute, 
or  I  ocus  of  centers  of  curvature,  is  an  equal 
cycloid.  To  construct  a  cycloid  (Fig.  66), 
divide  the  semi-circumference  of  the  gen- 
erating circle  into  n  equal  parts  (here  4) 
and  lay  off  these  arcs  along  the  base  (from 
O  to  4').  Describe  arcs  with  centers  at  1', 
2',  .  .  .  and  radii  equal  to  the  chords  Ol, 
O2,  .  .  . ,  and  sketch  the  cycloid  as  a  curve 
tangent  to  ail  of  these  arcs.  Or,  on  hori- 
zontal lines  through  1,2,.  .  .  lay  off  dis- 
tances equal  to  Ol',  02',  etc.;  the  points  thus  reached  will  lie  on  the 
cycloid. 

The  area  of  one  arch  =  37ra2,  length  of  arc  of  one  arch  =  8a.  Area 
bounded  by  the  ordinate  of  the  point  P  corresponding  to  any  value  of  u  is 
a2  ($$  rad  u  —  2  sin  u  +  J4  sin  2w)  =  %  ax  —  ft  y\/(2a  —y)y.  Length  of 
arc  OP  =  4a  (1  -  cos  #  w)  =  4a  -  2V/2a(2a  -  y). 


FIG.  67. 


152 


ANALYTICAL  GEOMETRY 


The  Trochoid  is  a  more  general  curve,  traced  by  any  point  on  a  radius 
of  the  rolling  circle,  at  distance  b  from  the  center  (Fig.  67).  It  is  a  prolate 
trochoid  if  6  <  a,  and  a  curtate  or  looped  trochoid  if  6  >  a.  The  equations  in 
either  case  are  x  =  a  rad  u  —  b  sin  u,  y  =  a  —  b  cos  w. 

The  Epicycloid  (or  Hypocycloid)  is  a  curve  generated  by  a  point  on  the 
circumference  of  a  circle  of  radius  a  which  rolls  without  slipping  on  the 
outside  (or  inside)  of  a  fixed  circle  of  radius  c.  For  the  equations,  put 
6  =  a  in  the  equations  of  the  epi-  (or  hypo-)  trochoid,  below.  The  normal 
at  any  point  P  passes  through  the  point  of  contact  N  of  the  corresponding 
position  of  the  rolling  circle.  To  construct  the  curve  (Figs.  68  and  69), 


Epicycloid. 
FIG.  68. 


Hypocycloid. 
FIG.  69. 


divide  the  semi-circumference  of  the  rolling  circle  into  n  equal  parts,  by  points 
1,  2,  3  .  .  . ,  and  lay  off  these  arcs  (Al,  A2,  A3)  along  the  circumference  of  the 
base  circle,  as  Al',  A2',  A3',  ....  Describe  circles  with  centers  at  1',  2',  3', 
.  .  .  and  radii  equal  to  the  chords  Al,  A2,  A3,  .  .  .;  then  the  required 
curve  will  be  tangent  to  all  these  circles.  Or,  with  0  as  center,  draw  arcs 
through  1,  2,  3,  .  .  .,  meeting  the  radius  OA  in  1°,  2°,  3°,  .  .  ..and  the 
radii  Ol',  O2',  03',  .  .  .  in  1",  2",  3",  .  .  .;  then  from  1",  2",  3",  .  .  . 
lay  off  arcs  equal  to  1°1,  2°2,  3°3,  .  .  .  respectively;  the  points  thus 
reached  will  be  points  of  the  curve. 


The  area  OAP  = 


a(c  ±  a)  (c  ±  2a) 
~2c 


(rad  u  —  sin  w) ,  where  the  upper  sign 


applies  to  the  epicycloid,  the  lower  to  the  hypocycloid,  and  rad  u  =  the 
radian  measure  of  the  angle  u  shown  in  Figs.  68  and  69.  Arc  AP  = 
(4  o/c)(c  ±  a)(l  -  cos  J4  w);  arc  AD  =  (4a/c)(c  ±  a).  [In  Fig.  69,  D  =4".] 

Radius  of  curvature  at  any  point  P  is  R  =  -^  '- —  sin  Yiu\  at  A,  R  =  0; 


c  ±2a 


at  £),  R 


4o(c  ±  o) 
c  ±  2a 


Special  Cases.     If  a  =  ^c,  the  hypocycloid  becomes  a  straight  line,  diam- 
eter of  the  fixed  circle  (Fig.  70).     In  this  case  the  hypo  trochoid  traced  by  any 


OTHER   USEFUL  CURVES 


153 


point  rigidly  connected  with  the  rolling  circle  (not  necessarily  on  the  circum- 
ference) will  be  an  ellipse.  If  a  =  He,  the  curve  generated  will  be  the  four- 
cusped  hypocycloid,  or  astroid,  (Fig.  71),  whose  equation  is  x^  +  y^  = 

c^.  If  a  =  c,  the  epicycloid  is  the  cardioid,  whose  equation  in  polar  co- 
ordinates (axes as  in  Fig.  72)  is  r  =  2c(l  +  cos  0).  Length  of  cardioid  =  16c. 


FIG.  70. 


Astroid. 
FIG.  71. 


Cardioid. 
FIG.  72. 


The  Epitrochoid  (or  Hypotrochoid)  is  a  curve  traced  by  any  point  rigidly 
attached  to  a  circle  of  radius  a,  at  distance  b  from  the  center,  when  this  circle 
rolls  without  slipping  on  the  outside  (or  inside)  of  a  fixed  circle  of  radius  c. 

The  equations  are  x  =  (c  ±  a)  cos  (  ^  u  J  +  6  cos     (  1  ±  ~  \  u  \, 

±d)  sin  \^-u)  —  &  sin     (  1  ±  —  J  u    .where  u  =  the  angle  which  the 


V  =  (c 

moving  radius  makes  with  the  line  of  centers;  take  the  upper  sign  for  the  epi- 
and  the  lower  for  the  hypo-trochoid.  The  curve  is  called  prolate  or  curtate 
according  as  6  <  a  or  6  >  a.  When  &  =  a,  the  special  case  of  the  epi-  or  hypo- 
cycloid  arises,  j 

The  Involute  of  a  Circle  is  the  curve  traced  by  the  end  of  a  taut  string 
which  is  unwound  from  the  circumference  of  a  fixed  circle,  of  radius  c.     If  QP 


Involute  of  Circle. 
FIG.  73. 


Spiral  of  Archimedes. 
FIG.   74. 


is  the  free  portion  of  the  string  at  any  instant  (Fig.  73),  QP  will  be  tangent  to 
the  circle  at  Q,  and  the  length  of  QP  =  length  of  arc  QA ;  hence  the  construe- 


154 


ANALYTICAL  GEOMETRY 


tion  of  the  curve.  The  equations  of  the  curve  in  parametric  form  (axes  as 
in  figure)  are  x  =  c(cos  u  +  rad  u  sin  w),  y  =  c  (sin  u  —  rad  u  cos  w), 
where  rad  u  is  the  radian  measure  of  the  angle  u  which  OQ  makes  with  the 
x-axis.  Length  of  arc  AP  =  ££c(rad  w)2;  radius  of  curvature  at  P  is  QP. 

The  Spiral  of  Archimedes  (Fig.  74)  is  traced  by  a  point  P  which,  starting 
from  O,  moves  with  uniform  velocity  along  a  ray  OP,  while  the  ray  itself 
revolves  with  uniform  angular  velocity  about  O.  Polar  equation:  r  = 
k  rad  0,  or  r  =  a  (0/360°).  Here  a  =  2irk  =  the  distance,  measured  along  a 
radius,  from  each  coil  to  the  next. 

In  order  to  construct  the  curve,  draw  radii  Ol,  O  2,  O3,  .    .    .  making  angles 

-  (360°),  -(360°),  -  (360°),  .  .  with  Ox,  and  along  these  radii  lay 
n  n  n 

123 

off  distances  equal  to  -  a,    -  a,    -  a,   . 
n        n        n 


the  points   thus   reached    will 


lie  on  the  spiral.  The  figure  shows  one-half  of  the  curve,  corresponding  to 
positive  values  of  6. 

Construction  for  tangent  and  normal:  Let  PT  and  PN  be  the  tangent 
and  normal  at  any  point  P,  the  line  TON  being  perpendicular  to  OP.  Then 
OT  =  r*/k,  and  ON  =  k,  where  k  =  a/(2ir).  Hence  the  construction. 

The  radius  of  curvature  at  P  is  R  =  (k2  +  r2)^/(2fc2  +  r2).  To  con- 
struct the  center  of  curvature,  C,  draw  NQ  perpendicular  to  PN  and  PQ 
perpendicular  to  OP;  then  OQ  will  meet  PN  in  C.  Length  of  arc  OP  = 
\$k  [rad  9  Vl  +  (rad  0)2  +  sinh-1(rad  6)].  After  many  windings,  arc  OP  = 
,  approximately. 


Hyperbolic  Spiral. 
FIG.  75. 


Logarithmic  Spiral. 
FIG.  76. 


The  Hyperbolic  Spiral  is  the  curve  whose  polar  equation  is  r  =  o/rad  0. 
To  construct  the  curve,  take  a  series  of  points  along  Ox  (Fig.  75) ;  through 
each  of  these  points,  with  center  at  O,  draw  an  arc  extending  into  the  upper  half 
of  the  plane;  and  along  each  of  these  arcs  lay  off  a  length  =  a.  The  points 
thus  reached  will  lie  on  the  curve.  A  line  parallel  to  the  x-axis,  at  distance  o, 
is  an  asymptote  of  the  curve.  The  curve  winds  around  and  around  the 
point  O  without  ever  reaching  it  (asymptotic  point) .  The  figure  shows  one- 
half  of  the  curve,  corresponding  to  positive  values  of  6.  HPT  and  PN  are  the 
tangent  and  normal  at  any  point  P,  the  line  TON  being  perpendicular  to  OP, 


OTHER   USEFUL  CURVES 


155 


then  OT  =  a,  and  ON  =  r2/a.  Hence  a  construction  for  the  tangent  and 
normal.  Radius  of  curvature  at  Pis  R  =  r/sin3  v,  where  v  =  angle  between 
OP  and  the  tangent  at  P.  Construction:  At  N  draw  a  perpendicular  to  PN, 
meeting  PO  in  Q; at  Q  draw  a  perpendicular  toPQ,  meeting  PN  in  C;  then  C  is 
the  center  of  curvature  for  the  point  P. 

The  Logarithmic  Spiral  (Fig.  76)  ,-is  a  curve  which  cuts  the  radii  from  O 
at  a  constant  angle  v,  whose  cotangent  is  m.  Polar  equation:  r  =  aem  fl. 
Here  a  is  the  value  of  r  when  9=0.  For  large  negative  values  of  0,  the  curve 
winds  around  O  as  an  asymptotic  point.  If  PT  and  PN  are  the  tangent  and 
normal  at  P,  the  line  TON  being  perpendicular  to  OP  (not  shown  in  fig.), 
then  ON  =  rm,  and  PN  =  r\A  +  w2  =  r/sin  v.  Radius  of  curvature  at 
P  is  PN.  The  evolute  of  the  spiral  is  an  equal  spiral 
whose  axis  makes  an  angle  ^TT  —  (loge  m)  /m  with  the 
axis  of  th'e  given  spiral.  Area  swept  out  by  the  radius 
r  from  r  =  0  (where  6  =  -co)  to  r  =  r,  is  A  = 
rV(4ra)  =  half  the  triangle  OPT.  Length  of  arc  from 
O  to  P  =  s  =  r/cos  v  =  PT. 

The  Tractrix,  or  Schiele's  Anti-friction  Curve 
(Fig.  77) ,  is  a  curve  such  that  the  portion  PT  of  the 
tangent  between  the  point  of  contact  and  the  z-axis  is 

constant  =  o.     Its    equation   is  x  =   +  a  I  cosh"1 


M         T 
Tractrix. 

FIG.  77. 


=   ±  a     ( 


or, 


in 


i  —  _ 

y 

parametric  form,  x  =  ±  a  [t  —  tanh  t],  y  =  a/cosh  t.  (For  tables  of  hyper- 
bolic functions,  see  p.  60.)  The  z-axis  is  an  asymptote  of  the  curve. 
Length  of  arc  BP  =  a  loge  (a/y}.  The  evolute  (locus  of  centers  of  curvature) 
is  the  catenary  whose  lowest  point  is  at  B,  and  whose  directrix  is  Ox. 

The  Cissoid  (Fig.  78)  is  the  locus  of  a  point  P  such  that  OP,  laid  off  on  a 
variable  ray  from  O,  is  equal  to  BD,  the  portion  of  the  ray  lying  between  a 
fixed  circle  through  O  and  a  fixed  tangent  at  the  point  A  opposite  O.  If  a 
is  the  radius  of  the  circle,  the  polar  equation  is  r  =  2a  sin2  6  /cos  6.  Rec- 
tangular equation,  y*(2a  —  x)  =  x3. 


Cissoid. 
FIG.  78. 

The  Lemniscate  (Fig.  79)  is  the  locus  of  a  point  P  the  product  of  whose 
distances  from  two  fixed  points  F,  Fr  is  constant,  equal  to  Yt  o2.  The  distance 
FFf  =  a^/2.  Polar  equation  is  r  =  a  A/cos  26.  Angle  between  OP  and  the 
normal  at  P  is  29.  The  two  branches  of  the  curve  cross  at  right  angles  at  O. 


156 


ANALYTICAL  GEOMETRY 


Zr  — H 


Maximum  y  occurs  when  9  =  30°  and  r  =  a/V^  and  is  equal  to  H  a 
Area  of  one  loop  =»  a2/2. 

The  Helix  (Fig.  80)  is  the  curve  of  a  screw  thread  on  a  cylinder  of  radius  r. 
The  curve  crosses  the  elements  of  the  cylinder  at  a  con- 
stant angle,  v.  The  pitch,  h,  is  the  distance  between  two 
coils  of  the  helix,  measured  along  an  element  of  the  cylinder  ; 
hence  h  =  2irr  tan  v.  Length  of  one  coil  =  \/(27rr)2  +  h2 
=  27rr/cos  v.  To  construct  the  projection  of  a  helix  on 
a  plane  containing  the  axis  of  the  cylinder,  draw  a  rectangle, 
breadth  2r  and  height  h,  to  represent  the  plane,  with  a 
semicircle  below  it,  as  in  the  figure,  to  represent  the  base 
of  the  cylinder.  Divide  h  into  equal  parts  (here  8),  num- 
bered from  1  to  8;  think  of  the  circumierence  as  also 
divided  into  8  equal  parts,  represented  on  the  semicircle 
by  numbers  from  1'  to  4'  and  back  again  from  4'  to  8'. 
Then  the  point  of  intersection  of  a  horizontal  line  through 
1,2,  .  .  .  with  a  vertical  line  through  1',  2',  ...  will 
be  a  point  of  the  required  projection.  If  the  cylinder  is 


X, 

^x^ 

X 

s 

2 

^s' 

^^ 

/ 

\Jr  & 

-*S* 

2' 

Helix. 

FIG.  80. 


rolled  out  on  a  plane,  the  development  of  the  helix  will  be  a  straight  line, 
with  slope  equal  to  tan  v. 


DIFFERENTIAL  AND  INTEGRAL  CALCULUS 

DERIVATIVES  AND  DIFFERENTIALS 

Derivatives  and  Differentials.  A  function  of  a  single  variable  x  may 
be  denoted  byf(x),F(x),  etc.  The  value  of  the  function  when  x  has  the  value 
XQ  is  then  denoted  by  f(xo),F(xo),  etc.  The  derivative  of  a  function  y  =  f(x) 
may  be  denoted  by  f'(x),  or  by  dy/dx.  The  value  of  the  derivative  at  a 
given  point  x  =  XQ  is  the  rate  of  change  of  the  function  at  that  point; 
or,  if  the  function  is  represented  by  a  curve  in  the  usual  way  (Fig.  1),  the 
value  of  the  derivative  at  any  point  shows  the  slope  of  the  curve  (that  is, 
the  slope  of  the  tangent  to  the  curve)  at  that  point 
(positive  if  the  tangent  points  upward,  and  negative 
if  it  points  downward,  moving  to  the  right) . 

The  increment, Ay  (read:  "delta  y"),  in  y  is  the 
change  produced  in  y  by  increasing  x  from  XQ  to  XQ  + 
Ax;  that  is,  Ay  =  f(x0  +  Ax)  -  /(z0).  The  differ- 
ential, dy,  of  y  is  the  value  which  Ay  would  have  if 
the  curve  coincided  with  its  tangent.  (The  differen- 
tial,  dx,  of  x  is  the  same  as  Ax  when  x  is  the  inde-  — 
pendent  variable.)  Note  that  the  derivative  depends 
only  on  the  value  of  XQ,  while  Ay  and  dy  depend  not  -^IG*  •• 

only  on  XQ  but  also  on  the  value  of  Ax.     The  ratio 

Ay /Ax  represents  the  slope  of  the  secant,  and  dy/dx  the  slope  of  the  tan- 
gent (see  Fig.  1).  If  Ax  is  made  to  approach  zero,  the  secant  approaches 
the  tangent  as  a  limiting  position,  so  that  the  derivative  =  f'(x)  = 


j  A  ,v.  -i-  rv  i    A       i          A /»  -i.  n  i  A-  i  •     "-~~>  My 

The  symbol  "lim"  in  connection  with  Ax  ==  0  means  "the  limit,  as  Ax 
approaches  0,  of  ..."  [A  constant  c  is  said  to  be  the  limit  of  a  variable  u  if, 
whenever  any  quantity  m  has  been  assigned,  there  is  a  stage  in  the  variation- 
process  beyond  which  |c  —  u\  is  always  less  than  m;  or,  briefly,  c  is  the  limit 
of  u  if  the  difference  between  c  and  u  can  be  made  to  become  and  remain  as 
small  as  we  please.] 

To  find  the  derivative  of  a  given  function  at  a  given  point:  (1)  If  the 
function  is  given  only  by  a  curve,  measure  graphically  the  slope  of  the 
tangent  at  the  point  in  question;  (2)  if  the  function  is  given  by  a  mathematical 
expression,  use  the  following  rules  for  differentiation.  These  rules  give, 
directly,  the  differential,  dy,  in  terms  of  dx\  to  find  the  derivative,  dy/dx, 
divide  through  by  dx. 

Rules  for  Differentiation.     (Here  u,v,w,.    .    .  represent  any  functions 
of  a  variable  x,  or  may  themselves  be  independent  variables,     a  is  a  constant 
which  does  not  change  in  value  in  the  same  discussion;  e  —  2.71828.) 
1.  d(a  +  u)   =  du.  2.  d(au)   =  adu. 

3.  d(u  +  v  +  w  +  .    .    .)  =  du  +  dv  +  dw  +  .    .    .      . 

4.  d(uv)   =  udv  +  vdu. 

.  (du    ,   dv       dw  \ 

5.  d(uvw .    .    . )   =  (uvw ...)( 1 1 h    •    •    .    ) 

\  u         v         w  I 

6    du  =  vdu  -udr,  t 

V  V2 

7.  d(um)   =  mum-ldu     when     m     is     not 

157 


158 


DIFFERENTIAL  AND  INTEGRAL  CALCULUS 


=  2wdu;  d(w3) 
_du_ 

2V 'u 

=  *?. 

u 

14.  d  sin  M  =  cos  udu. 
16.  d  cos  w  =   —  sin  udu. 
18.  d  tan  u  =  sec2  wdw. 

dw 
20.  d  sin-1  w 


Thus,  d(w2) 

,- 
8.  dVu  = 

10.  d(eM)   = 
12. 


3w2dw;  etc. 
9. 


11.  d(aw)   =  (log*  a)a"dw. 
13.  d  loglo  w  =  (logwe)  — 


u 


:  (0.4343.   .  )^. 


22.  d  cos-1  u  =  — 


15.  d  esc  M  =   —  cot  u  esc  u  du. 
17.  d  sec  u  =  tan  u  sec  w  du. 
19.  d  cot  w  =  —  esc2  w  du. 

du 
21.  d  csc-Jw  = - 


-  1 


d  sec-1  u  = 


dw 


24.  d  tan-1  w  =  „ 

1  +  w2 

26.  d  log«  sin  u  =  cot  w  du. 


25.  d  cot-1  u  =  — 


27.  d  log«    tan 


du 


1  +  w* 
2du 


sn 


28.  d  log«  cos  M  =   —  tan  u  du.  29.  d  log«  cot  u  =   — 


30.  d  sinh  w  =  cosh  u  du. 

32.  d  cosh  u  =  sinh  w  du. 

34.  d  tanh  w  =  sech2  u  du. 

36.  d  sinh-1  M  =  — f 


sin  2w 

31.  d  csch  w  =  —  csch  u  coth  wdu. 
33.  d  sech  u  =  —  sech  w  tanh  w  du. 
35.  d  coth  w  =  —  csch2  w  du. 


37.  d  csch-1  M  =  - 


+  1 


38.  d  cosh-1  M  = 


40.  d  tanh-1  u 


du 


\/u*  -  I 
du 


39.  d  sech-1  M  =  - 


dw 


41.  d 


1  -  u* 


1  -  u* 

42.  d(uv)   =  (uv~l)(u  logeu  dv  +vdu~). 

Derivatives  of  Higher  Orders.  The  derivative  of  the  derivative  ia 
called  the  second  derivative;  the  derivative  of  this,  the  third  derivative;  and 
BO  on.  Notation:  if  y  =  f(x), 


/'(*)  =  D,y 


dx' 


— ,;     etc. 


NOTE.    If  the  notation  fry/dx*  is  used,  this  must  not  be  treated  as  a  fraction,  like  dy/dx' 
but  as  an  inseperable  symbol,  made  up  of  a  symbol  of  operation,  d2/dx2,  and  an  operand  y 

The  geometric  meaning  of  the  second  derivative 
is  this:  if  the  original  function  y  =  f(x)  is  repre- 
sented by  a  curve  in  the  usual  way,  then  at  any 
point  where  f"(x)  is  positive,  the  curve  is  concave 
upward,  and  at  any  point  where  f"(x)  is  negative, 
the  curve  is  concave  downward  (Fig.  2).  When 
f"(x)  -  0,  the  curve  usually  has  a  point  of 
inflection.  Fio.  2. 

Differentials  of  Higher  Orders.  The  differ- 
ential of  the  differential  is  called  the  second  differential;  the  differential  of 


DERIVATIVES  AND  DIFFERENTIALS;  MAXIMA  AND  MINIMA     159 

this,  the  third  differential;  etc.     These  quantities  are  of  little  importance 
except  in  the  case  where  dx  =  a  constant.     In  this  case 

dy=f(x-)dx;    d*y  =/"(*)•(<**)»;     dzy  =  /'"(*)  -(dx)z;   .    .     . 
The  first,  second,  third,  etc.,  differentials  are  close  approximations  to  the  first, 
second,  third,  etc.,  differences  (p.  115),  and  are  therefore  sometimes  useful  in 
constructing  tables.     Thus,  denoting  the  first,  second,  third,  etc.,  differences  by 
£>',  D",  D'",  etc.,  and,  assuming  always  that  dx  —  a  constant, 

£'     =dy+Kd*y  +Kd*y  +  Kidiy  +  .    .    .  ;  d*y  =  D>"  -  %  D""  +  .    .'    . 

D"    =  d*y  +  d3y  +  Vi2  d*y  +  .    .    .  ;  dzy  =  D"  -  D"'  +  itfa  D""  +  .    .    . 

D'"  =  d*y  +  %  d*y  +  .    .    .  ;  dy  =  D'  -  H  D"  +  H  D'"  -  H  D""  +  .    .    . 

Functions  of  Two  or  More  Variables  may  be  denoted  by  f(x,  y,  .  .  .  ), 
F(x,  y,  .  .  .),etc.  The  derivative  of  such  a  function  u  =  f(x,  y,  .  .  .)  formed 
on  the  assumption  that  x  is  the  only  variable  (y,  .  .  .  being  regarded  for  the 
moment  as  constants)  is  called  the  partial  derivative  of  u  with  respect  to 

x,    and  is  denoted  by  fx(x,y),  or  Dxu,  or  -^-,  or  ^--     Similarly,  the  partial 

dx          ox 

derivative  of  u  with    respect  to  y   is  fv(x,y),  or  Dvu,  or  ——,  or  -r-- 

dy         dy 

NOTE.  In  the  third  notation,  dxu  denotes  the  differential  of  u  formed  on  the  assump- 
tion that  x  is  the  only  variable.  If  the  fourth  notation,  du/'dx,  is  used,  this  must  not  be 
treated  as  a  fraction  like  du/dx;  the  d/dx  is  a  symbol  of  operation,  operating  on  u,  and 
the  "  dx  "  must  not  be  separated. 

Partial  derivatives  of  the  second  order  are  denoted  byfxx,  fxu,  fvv,  or  by  D\u, 

-^  —  -'  ,    ..    J 
oxz    oxoy 

Similarly  for  higher  derivatives.     Note  that/xj,  =  fvx. 

If  increments  A#,  Ay,  (or  dx,  dy)  are  assigned  to  the  independent  variables 
x,  y,  the  increment,  Aw,  produced  in  u  =  f(x,y)  is 

AM  =  f(x  +  Az,  y  +  Ay)   -  f(x,y)  ; 

while  the  differential,  du,  that  is,  the  value  which  Aw  would  have  if  the  partial 
derivatives  of  u  with  respect  to  x  and  y  were  constant,  is  given  by 

du  =  (fj-dx  +Vv)'dy. 

Here  the  coefficients  of  dx  and  dy  are  the  values  of  the  partial  derivatives  of  u 
at  the  point  in  question. 

If  x  and  y  are  functions  of  a  third  variable  t,  then  the  equation 


expresses  the  rate  of  change  of  u  with  respect  to  t,  in  terms  of  the  separate  rates 
of  change  of  x  and  y  with  respect  to  t. 

For  the  graphical  representation  of  u  =  f(x,y),  see  p.  178. 

Implicit  Functions.  If  f(x,y)  =  0,  either  of  the  variables  x  and  y  is 
said  to  be  an  implicit  function  of  the  other.  To  find  dy/dx,  either  (1)  solve 
for  y  in  terms  of  x,  and  then  find  dy/dx  directly;  or  (2)  differentiate  the  equa- 
tion through  as  it  stands,  remembering  that  both  x  and  y  are  variables,  and 
then  divide  by  dx;  or  (3)  use  the  formula  dy/dx  =  —  (/*//,/),  where  fx  and 
fy  are  the  partial  derivatives  of  f(x,y~)  at  the  point  in  question. 

MAXIMA  AND  MINIMA 

A  Function  of  One  Variable,  as  y  =  f(x),  is  said  to  have  a  maximum  at 
a  point  x  =  x<\,  if  at  that  point  the  slope  of  the  curve  is  zero  and  the  concavity 


Dx(Duu),  Dlu,  or  by  -^  —  -'  ,    ..    J  -^  —  -'  the  last  symbols  being  "inseparable." 


160 


DIFFERENTIAL  AND  INTEGRAL  CALCULUS 


FIG.  3. 


downward  (see  Fig.  3) ;  a  sufficient  condition  for  a  maximum  is  f(xo)  =  0 
and  /"(ZQ)  negative.  Similarly,  /(z)  has  a  minimum  if  the  slope  is  zero  and 
the  concavity  upward;  a  sufficient  condition  for  a  minimum  is  /'(x0)  =  0  and 
/"(XQ)  positive.  If  /'(#o)  =  0  and  f'f(xo)  = 
0  and  f"(xo)  7*  0,  the  point  XQ  will  be  a 
point  of  inflection.  If  f(x0)  =  0  and 
f"(xo)  =  0  and  f'"(xo)  =  0,  the  point  x0will 
be  a  maximum  if  /""(XQ)  <  0,  and  a  mini- 
mum if  f""(xo)  >  0.  It  is  usually  sufficient, 
however,  in  any  practical  case,  to  find  the 
values  of  x  which  make  f(x)  =  0,  and  then 
decide,  from  a  general  knowledge  of  the 
curve,  which  of  these  values  (if  any)  give 
maxima  or  minima,  without  investigating  the  higher  derivatives. 

A  Function  of  Two  Variables,  as  u  =  f(x-,y),  will  have  a  maximum 
at  a  point  (x0,y0)  if  at  that  point  /*  =  0,  fy  =  0,  and  fxx  <  0,  fvv  <  0 ; 
and  a  minimum  if  at  that  point  fx  =  0,  fu  =  0,  and  fxx  >  0,  fyy  >  0; 
provided,  in  each  case,  (/**)  (fyv)  —  (fxy} 2  is  positive.  If  /*  =0  and  fv  =  0, 
and  fxx  and  fyyhave  opposite  signs,  the  point  (#0,2/0)  will  be  a  "saddle  point" 
of  the  surface  representing  the  function  (p.  178). 

EXPANSION  IN  SERIES 

The  range  of  values  of  x  for  which  each  of  the  series  is  convergent  is  stated 
at  the  right  of  the  series. 

Arithmetical  and  Geometrical  Series,  and  the  Binomial  Theorem. 
See  p.  114. 

Exponential  and  Logarithmic  Series. 


a>0,    — 


where  ra  =  log*  a  =  (2.3026)  (logio  a). 


x        x        x 


X*          Or*          X*          X* 


-  1  <x  <  +1. 


-  l<x<  +1. 

x<  -  lor  +  1  <x. 

0  <  X  <    oo. 


log.(a+*) 


0  <a  <  4- 
-a  <x  <  + 


EXPANSION  IN  SERIES  161 

Series  for  the  Trigonometric  Functions.  In  the  following  formulae, 
all  angles  must  be  expressed  in  radians.  If  D  —  the  number  of  degrees  in 
the  angle,  and  x  =  its  radian  measure,  then  x  =  0.017453  D. 

,      X3          X6          X7 

sin*=*--  +  --^  +  ...;  =><  x  <  +  oo. 

Xa          X*          X6          X8 

C08I  =  1--+---+--.  .  ..  -.<,<  +  .. 

x3   2x6   17x7   62x9 

' 


1       x      x3       2x6         x1 
cot*-------—-—-.    .    .; 


tan-1  y  =  y-£-  +  £--£-  +  ...;  -  1  £  y  £  +  1. 

o  O  / 

cos"1  y  =  H?r  —  sin~'  y;        cot"1  y  =  ^iir  —  tan—1  y. 

Series  for  the  Hyperbolic  Functions  (x  a  pure  number). 

X3          X6          X7 

sinh  x==x+^7  +  £7  +  7J  +  ---;  -eo<z<o>. 

X2          X4          X6 

coshx  =  1  +—  +  —+—  +  .    .    .;  -oo<x<oo. 


General  Formulae  of  Maclaurin  and  Taylor.  If  /(x)  and  all  its  deriva- 
tives are  continuous  in  the  neighborhood  of  the  point  x  =  0  (or  x  =  a)  ,  then, 
for  any  value  of  x  in  this  neighborhood,  the  function  /(x)  may  be  expressed 
as  a  power  series  arranged  according  to  ascending  powers  of  x  (or  of  x  —  o), 
as  follows: 


.  .  . 

+  x"-1  +  (Pn)x".      (Maclaurin.) 


+    (n  _         (x  -  a)"'1  +  (Qn)(x  -  a)".      (Taylor.) 

Here  (Pn)xn,  or  (Qn)(x  —  a)n,  is  called  the  remainder  term  ;  the  values  of 
the  coefficients  Pn  and  Qn  may  be  expressed  as  follows: 
Pn  =  {/(")(sx)}/n!  =  {(1  -  O"-1    fW(tx)}/(n-  1)1 
Qn  =  {/C">[a  +  s(x  -  a)]}/n!  =  {(1  -  Ow~l  /<">[<*  +  t(x  -  a)]}/(n  -  1)! 
where  s  and  t  are  certain  unknown  numbers  between  0  and  1  ;  the  s-form  is 
due  to  Lagrange,  the  /-form  to  Cauchy. 

The  error  due  to  neglecting  the  remainder  term  is  less  than   (Pn)xn,   or 


162 


DIFFERENTIAL  AND  INTEGRAL  CALCULUS 


(Qn)(x  ~  °)n»  where  Pn,  or  Qn,  is  the  largest  value  taken  on  by  Pn,  or  Qn, 
when  s  or  t  ranges  from  0  to  1.  If  this  error,  which  depends  on  both  n  and 
x,  approaches  0  as  n  increases  (for  any  given  value  of  x)  ,  then  the  general  - 
expression-with-remainder  becomes  (for  that  value  of  x)  a  convergent  in- 
finite series. 

The  sum  of  the  first  few  terms  of  Maclaurin's'series  gives  a  good  approxi- 
mation to  f(x)  for  values  of  x  near  x  =  0;  Taylor's  series  gives  a  similar  ap- 
proximation for  values  near  x  =  a. 

Fourier's  Series.  Let  f(x)  be  a  function  which  is  finite  in  the  interval 
from  x  =  —  c  to  x  =  +  c  and  has  only  a  finite  number  of  discontinuities  in 
that  interval  (see  note  below),  and  only  a  finite  number  of  maxima  and 
minima.  Then,  for  any  value  of  x  between  —  c  and  c, 


f(x~) 


00+ 


cos  —  +  02  cos  ---  \-  as  cos  -  --  \- 
c  c  c 


.       TTX      ,  .       2WX  .       3-7TX 

+  &i  sin  —  +62  sin  -  •   +  &3  sin  -- 
c  c  c 

where  the  constant  coefficients  are  determined  as  follows: 


/(0COS  —  dl, 

C 


/(<)  sin—  dt. 
—  c 


In  case  the  curve  y  =  f(x)  is  symmetrical  with  respect  to  the  origin,  the 
a's  are  all  zero,  and  the  series  is  a  sine  series.  In  case  the  curve  is  sym- 
metrical with  respect  to  the  y-axis,  the  6's  are  all  zero,  and  a  cosine  series 
results.  (In  this  case,  the  series  will  be  valid  not  only  for  values  of  x  between 
—  c  and  c,  but  also  for  x  =  —  c  and  x  —  c.)  A  Fourier's  series  can  be  inte- 
grated term  by  term;  but  the  result  of  differentiating  term  by  term  will  in 
general  not  be  a  convergent  series. 

NOTE.  If  x  =  xo  is  a  point  of  discontinuity,  f(xo)  is  to  be  defined  as  ^[fi(xo)  +  /2(xo)], 
where  fi(xo)  is  the  limit  of  f(x)  when  x  approaches  xo  from  below,  and  MXO)  is  the  limit 
of  f(x)  when  x  approaches  xo  from  above. 


^N- 


FIG.  4. 


o» 


FIG.  5. 


•H*-' 


/*    / 

/     I         /    I      * 


FIG.  6. 


Examples  of  Fourier's  Series. 

1.  If  y  =  f(x)  is  the  curve  in  Fig.  4, 

_  h  _  4A  /       THC        1        STI 
V  —  2  ~  K*  \COS  c         9C°S    c 

2.  If  i/  =  /(x)  is  the  curve  in  Fig.  5, 

4h    I    .     TTX       1     .      Zirx 


1         5irx  , 
25C°S^-  + 


5irx 


3.  If  y  =  /(x)  is  the  curve  in  Fig.  6, 

2h     I    .       TTX  1      .       27TZ 


y  =~     8m- 


INDETERMINATE  FORMS;  CURVATURE  163 

% 
INDETERMINATE  FORMS 

In  the  following  paragraphs,  f(x),  g(x)  denote  functions  which  approach  0; 
F(x),G(x)  functions  which  increase  indefinitely;  and  U(x)  a  function  which  ap- 
proaches 1  ;  when  x  approaches  a  definite  quantity  a.  The  problem  in  each 
case  is  to  find  the  limit  approached  by  certain  combinations  of  these  functions 
when  x  approaches  a.  The  symbol  =  is  to  be  read  "approaches." 

CASE  1.     "jj-"     To  find  the  limit  of  f(x)/g(x)  when  /(x)  =  0  and0(z)  =  0, 

use   the  theorem   that   lim  —  —  =  lim  »  where  f'(x)   and   g'(x)  are  the 

derivatives  of  f(x)  and  g(x).     This  second  limit  may  be  easier  to  find  than  the 
first.      If  f'(x)  =  0  and  g'(x)  =  0,  apply  the  same  theorem  a  second  time: 

Hm^~  "  =  lim  ^7^;  and  so  on. 
g'(x)  Q"(X) 

CASE  2.     "-^-"     If  F(x)  =  oo   and(?(z)  =  oo,  then  li 

precisely  as  in  Case  1. 

CASES.  "O-oo."  To  find  the  limit  of  f(x)  -F(x)  when  f(x)  =  0  and  F(x)  =  oo, 

write   lim    [f(x)-F(x)]  =  lim  ,      *  s  ,  or  =  lim  ^  '  ,  .  ;    then    proceed    as    in 


Case  1  or  Case  2. 

CASE  4.     "0°."     If  /(*)  =  0  and  g(x)  =  0,  find   lim  [/(x)f(x)  as  follows: 
let    y   =  [f(x)}°^,  and  take  the  logarithm  of  both   sides  thus: 

log«  y  =  (/(x)  loge  /(x)  ; 
next,  find  lim  [g(x)  loge/(x)],   =  m,  by  Case  3;  then  lim  y  =  em. 

CASE  5.     "1°°."     If  U(x)  =   1  andF(x)  =  oo,  find  lim  [C7(x)]^(x)as  follows: 
let  y  =    [U(x)\  F^x\  and  take  the  logarithm  of  both  sides,  as  in  Case  4. 

CASE  6.     "  coo."     if  p(x)  =  ooand  f(x)  =  0,  find  lim  [F(x)]f(x)  as  follows: 
let  y  =  [F(x)]f(x\  and  take  the  logarithm  of  both  sides,  as  in  Case  4. 

CASE  7.     "  oo-  oo."     If  F(x)  =  oo  and  G(x)  =  oo,  write  lim  [F(z)  —  O(x)] 
1  1 

=  lim  —   --   —  ;  then  proceed  as  in  Case  1.     Sometimes  it  is  shorter  to  ex- 

F(x)  -G(x) 

pand  the  functions  in  series.     It  should  be  carefully  noticed  that  expressions 
like  0/0,  o0/00.  etc.,  do  not  represent  mathematical  quantities. 

CURVATURE 

The  radius  of  curvature  R  of  a  plane  curve  at  any  point  P  (Fig.  7)  is  the 
distance,  measured  along  the  normal,  on  the  concave  side  of 
the  curve,  to  the  center  of  curvature,  C,  this  point  being 
the  limiting  position  of  the  point  of  intersection  of  the  nor- 
mals at  P  and  a  neighboring  point  Q,  as  Q  is  made  to  ap- 
proach P  along  the  curve.  If  the  equation  of  the  curve  ia 
V  =/(*), 

ds    _  [1  +  (yO']?* 
~  du  ~  g"  FIG.  7. 


154  DIFFERENTIAL  AND  INTEGRAL  CALCULUS 


where  ds  =  -\/dx2  +  dy2  =  the  differential  of  arc,  u  =  tan"1  [/'(a;)]  =  the 
a,ngle  which  the  tangent  at  P  makes  with  the  a>axis,  and  yf  =  f  (x)  and 
y"  =  /"  (x)  are  the  first  and  second  derivatives  of  f(x)  at  the  point  P.  Note 
that  dx  —  ds  cos  u  and  dy  =  ds  sin  u.  The  curvature,  K,  at  the  point  P,  is 
K  =  1/R  =  du/ds;  that  is,  the  curvature  is  the  rate  at  which  the  angle  u  is 
changing  with  respect  to  the  length  of  arc  s.  If  the  slope  of  the  curve  is  small, 
**•/»<*). 

If  the  equation  of  the  curve  in  polar  co-ordinates  is  r  =  /(0)  ,  where  r  =  radius 
vector  and  6  =  polar  angle,  then 


r2  -  rr"  +  2(r')2 
where  r'  =  /'(*)  and  r"  =/"(0). 

The  evolute  of  a  curve  is  the  locus  of  its  centers  of  curvature.  If  one  curve 
is  the  evolute  of  another,  the  second  is  called  the  involute  of  the  first. 

INDEFINITE  INTEGRALS 

An  integral  of  f(x)dx  is  any  function  whose  differential  iaf(x)dx,  and  is 
denoted  by  J"f(x)dx.  All  the  integrals  of  f(x)dx  are  included  in  the  ex- 
pression y*/(30ffcc+  C,  where  tff(x)dx  is  any  particular  integral,  and  C  is  an 
arbitrary  constant.  The  process  of  finding  (when  possible)  an  integral  of  a 
given  function  consists  in  recognizing  by  inspection  a  function  which,  when 
differentiated,  will  produce  the  given  function;  or  in  transforming  the  given 
function  into  a  form  in  which  such  recognition  is  easy.  The  most  common 
integrable  forms  are  collected  in  the  following  brief  table;  for  a  more  extended 
list,  see  B.  O.  Peirce's  "Table  of  Integrals"  (Ginn  &  Co.). 

GENERAL,  FORMULAE 

1.  fadu  =  afdu  =  au  +  C  2.  J*  (u  +  v)dx  =  fudx  -j-  fvdx 
3.  fudv  =  uv  -f-odu  4.  fj(x)dx  =  f  f[F(y}}Ff(y)dy,  x  =  F(y) 
5-  fdyff(x,y}dx  =  f  dxf  f(x,y}dy. 

FUNDAMENTAL  INTEGRALS 
_  xn+i 

6.  Jxndx  =  —  —    +  C;  when  n  ^   -  1 

7.  y  —   =  log«  x  +  C  =  loge  ex          8.   J*exdx  =  ex  +  C 

9.   ^sin  xdx  =  —  cos  x  +  C  10.   ^cos  xdx  =  sin  x  +  C 

tan  x  +  C 


tan'1  x  +  C  =   -  cot-1  x  +  c 


RATIONAL  FUNCTIONS 

_L  ki^n+1 


16. 
17. 
19. 
20. 


INDEFINITE  INTEGRALS 
1 

dx 


1-65 


=  loge  (a  +  bx} 


is. 


(a 


6(0 


dx 


loge 


\-C  =  tanh"1*  +C, 
+  C  =   -  coth-1  x  +  C, 


24   r      dx       = - 

J  a  +  2bx,+  ex2  b  + 


when  x  <  1 


when  z  >  1 


when  o  >  0,     6  >  0 


when 
ac  -  62  >  0 ; 


when 
&»  -  ac  >  0; 


dx 


A+ 


26. 


27. 


,  if  /(x)  is  a  polynominal  of  higher  than  the  first 

degree,  divide  by  the  denominator  before  integrating. 
dx  1  6  +  ex 


(a  +  2bx 


2(ac  —  bz)(p  —  1)      (a 
(2p  -  3)c 


28-^(^f^ 


nx)dx 


2(oc  - 
n 


-  1)  **    (o  +  2bx 
I 


2c(p  -  1)  "(a  +2bx  + 

me  —  rib    s*  dx 

~c        ^  (a  +  2bx  + 


29. 


+bx)ndx 


(m 


dx 


166 


DIFFERENTIAL  AND  INTEGRAL  CALCULUS 
IRRATIONAL  FUNCTIONS 

+  C 


-1  —  +  C  =   -  cos-1—  +  c 
a  a 


'a  +  bx,  and  use  21  and  22 


.  =  log*  [x  +  Va2  +  x2]  +C=  sinh-1  —  +  c 
=  loge  [x  +  V*z  -  a2]  +  C  =  cosh"1  -  +  c 


38.  y 


1 

+  2bx  +  ex2        Vc 

when  c  >  0; 

=  ^-  sinh"1    b+^x=  +  C,          when  ac  -62  >  0; 
7  tc  -62 

zcosh-1-^^  +  C,       when  62  -  ac  >  0; 
=  -^Lsin'1     b+CX     +  C,  when  c  <  0 


39.  y 


40.  y 


- V«2  +  #2  +  —  sinh   x he 

a 


INDEFINITE  INTEGRALS 


167 


-  a2  da;  =     V*2  -  a2  -        loge  (x  +  V*2  -  a2)  +  C 


44. 


TRANSCENDENTAL  FUNCTIONS 


45. 


46.  y 


ax  a2x2 

47.   y  loge  xdx  =  x  loge  x  —  x  +  C 


48. 


49. 


71    —p 


_      +  c 

x 


50.  y*sin2  xdx  =   —  \i  sin  2x  +  \to  +  C  =  —  M  sin  x  cos  x  +  Ha;  +  C 

51.  y*cos2  x  dx  =  J4  sin  2x  +  H£  +  C  =  %  sin  x  cos  x  +  Hx  -+-  C 

/>  .  cos  mx    ,  /»  sin  mx 

52.  /  sm  mx  dx  = h  C  53.     /  cos  mx  dx  =  — 1-  C 

m  m 

/»  .  cos  (m  +  n)x       cos  (m  —  ri)x 

54.  /  sm  mx  cos  nx  dx  = f-  C 

2(m  +  n)  2(m  —  n) 

/^  .  sin(m  —  n)x       sin(m  +  n)x    , 

55.  /  sm  mx  sin  nx  dx  =  —  7 '- \-  C 

J  2(m  -  n)  2(m  +  n) 

Kc       p  sin(m  —  ri)x       sin(m  -f-  ri)x    , 

oo.     /  cos  mx  cos  nx  ax  = -4-  C 

2(m  —  n)  2(m  +  n) 

57.   J  tan  xdx  =   —  log«  cos  x  +  C  58.   y*cot  xdx  =  log«  sin  x  +  C 

59.     f-     -  =  loge  tan  ~  +  C  60.     /* — —   =  log«  tan   (  -  +  -  )    +  < 

sm  x  2  *^    cos  x  \4       2/ 


63.     /'sin  x  cos  a; da;  =  ^  sin2  x  +  C     64.     /"- 

*X  t-'        ai 


—  cos  x 
dx 


sin  x  cos  x 


-  cot  -  +  C 

=  log,  tan  x  +  C 


65.*y*sinn  xdx  =   — 


66.*y*cosn  xdx  = 


sin  x  cos" 
n 


,   *  -1 
n  -  \    f 


2  x  dx 


n 


•  If  n  is  an  odd  number,  substitute  cos  x  =-  z  or  sin  a;  =•  z. 


168 
67. 


DIFFERENTIAL  AND  INTEGRAL  CALCULUS 


tan' 


~z  xdx 


/»                              cotn   l  x          r> 
68.   J  cot*  xdx  = _ J  cotn~2  xdx 


69.     r-*L_  = 

J     sin*  -r  fw.    — 


~2 


(n 


n  ~       r      <* 

n    —   I**      gin""2 


TO.  y 


dx 


sin"  x 

dx 

cosn  x       (n  —  1)  cos"""1  x       n  —  1«^   cosn~2  x 
sinp+1xcos9~1x 


k   r  . 

71.   J  smp  x  cos9  xdx  = 


4 /*sinpxcos«-Jxdx 

P  +  3  P  +  <T 


+ 


-  C sinp   2x  cos' xdx 
79  ~i~  C 


sinp+1xcos~g+1x         q  -  p  -2 
q  -  I  q  -  I      ' 


74. 


a  +  o  cos  a; 


2  /      a 

5^5    -Wi 


tan  *** 


sinp  a;  cos- 
C'  when  °s 


6  +  a  cos  x  4-  sin #v&2  —  a2 


a  4  b  cos  x 


,    2         tanh"1  (  \/^— — ^  tan  \te  )    4  C, 
&2_0a  \\64a  / 

_  x        a    r> 

o  4  6  cos  x        &        &  J  a  4  6  cos  x 


when 
a2  <  6s 


^»    cos  x  dx      _  x        a    r>         dx 

75'     J    n    0-  h  ™o  -r    =   T  ~     fc  »/    /i    4-  7>  nna  -r   "*" 


76.  r  * 

**    a  4- 

^»A  + 

77.      /    ; 

47     a+ 


=  —  —  log.  (a  4  b  cos  x)  4  C 

0  COS  X  0 


B  cos  x  4  C  sin  x  ,  .    /» 

: dx  =  A  I  

6  cos  x  4  c  sin  x  **   a  4 

cos  y  dy 


dy 


pcosy 


a  +  p  cos  y 
where  b  =  p  cos  w,  c  =  p  sin  w  and  z  —  M 

a  sin  bx  —  b  cos  & 
78. 


79. 

80. 
81. 
82. 
83. 


sin  y  dy 
a  4  pcosy' 


war1  xdx  =  xsin-1^  +       l  -  &  +  C 
cosT1  xdx  =  xcos~lx  —  \/l  —  x2  +  C 
tan"1  xdx  =  x  tan"1  x  -  J4  loge  (1  +  x2)  4-  C 
cot'^dx  =  x  cot"1  x  +  ^log«  (1  4  za)  4  C 
•  If  p  or  g  is  an  odd  number,  substitute  eos  z  •=  z  or  einx  —  ». 


DEFINITE  INTEGRALS  169 

84.  y*  sinh  xdx  =  cosh  x  +  C  85.     y  tanh  xdx  =  log,  cosh  a:  +  C 

86.  y  cosh  x  dx  =  sinh  a;  +  C  87.     fcoth  xdx  =  loga  sinh  x  +  C 

88.  y  sech  xdx  =  2  tan"1  (e*)  +  C  89.     fcach  xdx  =  loga  tanh  (x/2)  +C 

90.  A*  sinh2  xdx  =  ^  sinh  a:  cosh  x  —  MX  +  C 

91.  f  cosh2  xdx  =  \<i  sinh  at  cosh  x  +  ftx  +  C 

92.  y  sech2  xdx  =  tanh  x  +  C          93.     J*  csch2  xdx  =   -  coth  x  +  C 

DEFINITE  INTEGRALS 

The  definite  integral  of  /(x)dx  from  x  =  o  to  x  =  6,  denoted  by  ^  f(x)dx, 
is  the  limit  (as  n  increases  indefinitely)  of  a  sum  of  n  terms:    . 

fabf(x~)dx  =    ^  [/(xi)Ax  +/(x2)Ax  +/(x3)Ax  +  .    .    .   +  /(xn)Ax], 

built  up  as  follows:  Divide  the  interval  from  a  to  6  into  n  equal  parts,  and  call 
each  part  Ax,  =  (b  —  a)/n;  in  each  of  these  intervals  take  a  value  of  x  (say 
xi,  X2,  .  .  .  xn),  find  the  value  of  the  function  /(x)  at  each  of  these  points, 
and  multiply  it  by  Ax,  the  width  of  the  interval;  then  take  the  limit  of  the  sum 
of  the  terms  thus  formed,  when  the  number  of  terms  increases  indefinitely, 
while  each  individual  term  approaches  zero. 

Geometrically,  J^*f(x}dx  is  the  area  bounded  by  the  curve  y  =  /(x),  the 
x-axis,  and  the  ordinates  x  =  a  and  x  =  6  (Fig.  8)  ;  that 
is,  briefly,  the  "area  under  the  curve,  from  a  to  b."     The 
fundamental  theorem  for  the  evaluation  of  a  definite 
integral  is  the  following: 


that  is,  the  definite  integral  is  equal  to  the  difference  be- 
tween two  values  of  any  one  of  the  indefinite  integrals 
of  the  function  in  question.     In  other  words,  the  limit  of  a  sum  can  be  found 
whenever  the  function  can  be  integrated. 

Properties  of  Definite  Integrals. 


THE  MEAN-VALUE  THEOREM  FOR  INTEGRALS. 

f*F(.x)f(x)dx    =  F(X)  f*  f(x)dx, 

provided  /(x)  does  not  change  sign  from  x  =  a  to  x  =  b;  here  X  is  some  (un- 
known) value  of  x  intermediate  between  a  and  b. 

THEOREM  ON  CHANGE  OF  VARIABLE.  In  evaluating  J*x'  a  f(x)dx,  f(x)dx 
may  be  replaced  by  its  value  in  terms  of  a  new  variable  t  and  dt,  and  x  —  a 
and  x  =  b  by  the  corresponding  values  of  t,  provided  that  throughout  the 
interval  the  relation  between  x  and  t  is  a  one-to-one  correspondence  (that  is, 
to  each  value  of  x  there  corresponds  one  and  only  one  value  of  t,  and  to  each 
value  of  t  there  corresponds  one  and  only  one  value  of  x). 


170  DIFFERENTIAL  AND  INTEGRAL  CALCULUS 

DIFFERENTIATION  WITH  RESPECT  TO  THE  UPPER  LIMIT.     If  b  is  variable, 
then  J     f(x)dx  is  a  function  of  6,  whose  derivative  is 


DIFFERENTIATION  WITH  RESPECT  TO  A  PARAMETER. 


Functions  Denned  by  Definite  Integrals.  The  following  definite 
integrals  have  received  special  names,  and  their  values  have  been  tabulated  ; 
see,  for  example,  B.  O.  Peirce's  "Table  of  Integrals." 

dx 


, 
k2  sin2  x 


(*•<!) 


1.  Elliptic  integral  of  the  first  kind  *=F(u,K)  =  J]"     , — 

2.  Elliptic  integral  of  the  second  kind  =  E(u,  /b)   =  t/^"V'l  —  k2  sin2x  dx 
(k2  <  1) 

3.  4.   Complete  elliptic  integrals  of  the  first  and  second  kinds;  put  u  =  ir/2 
in  (1)  and  (2). 

5.  The  Probability  integral  =  —j=-J**e~x'idx 

6.  The  Gamma  function  =  F(n)   =  f^  m  xn~1e~xdx 

Approximate  Methods   of  Integration.     Mechanical    Quadrature. 

(1)  Use  Simpson's  rule.     See  p.  106. 

(2)  Expand  the  function  in  a  power  -series,  and  integrate  term  by  term. 

(3)  Plot  the  area  under  the  curve  y  =  f(x)  from  x  =  a  to  x  =  b  on  squared 
paper  and  measure  this  area  roughly  by  "counting  squares,"  or  more  accu- 
rately, by  the  use  of  a  planimeter  ($14  to  $35;  instruction  for  use  with  each 
instrument). 

(4)  Coradi's  Mechanical  Integraph  ($240)  provides  a  means  of  drawing 
on  paper  the  curve  y  =  ff(x}dx,  when  the  curve  y  =  f(x~)  is  given,  and  can 
be  used  to   facilitate  the  solution  of   certain  differential   equations.     Full 
instructions  for  use  with  each  instrument. 

Double  Integrals.  The  notation  fff(x,  y)dy  dx 
means  J"  \f  f(x,  y)dy}dx,  the  limits  of  integration  in  the 
inner,  or  first,  integral  being  functions  of  x  (or  constants). 

EXAMPLE.  To  find  the  weight  of  a  plane  area  whose 
density,  w,  is  variable,  say  w  =  f(x,  y).  The  weight  of  a 
typical  element,  dx  dy,  is  f(x,  y)dx  dy.  Keeping  x  and 
dx  constant,  and  summing  these  elements  from,  say,  y  —  jrIGt  g. 

Fi(x)  to  y  =  F*t(x),  as  determined  by  the  shape  of  the 

boundary,    the    weight  of    a    typical  strip    perpendicular    to  the    £-axis  is 
fy=Fz(x) 

dx     I  f(x,y)dy.     Finally,  summing  these  strips  from,  say,  x  —  a  to  x  =  b,  the 


Jy=Fi(x) 

fx  =  b    /™y=Fz(x) 

weight  of  the  whole  area  is    I    {dx    I   f(x,  y}dy] ,  or,  briefly,  f  J*f(x,  y)dydx. 
Jx  =  a    Jy=Fi(x) 


DIFFERENTIAL  EQUATIONS  171 

DIFFERENTIAL  EQUATIONS 

An  ordinary  differential  equation  is  one  which  contains  a  single  inde- 
pendent variable,  or  argument,  and  a  single  dependent  variable,  or  function, 
with  its  derivatives  of  various  orders.  A  partial  differential  equation  is 

one  which  contains  a  function  of  several  independent  variables,  and  its  partial 
derivatives  of  various  orders.  The  order  of  a  differential  equation  is  the  order 
of  the  highest  derivative  which  occurs  in  it.  A  solution  of  a  differential 
equation  is  any  relation  between  the  variables,  which,  when  substituted  in 
the  given  equation,  will  satisfy  it.  The  general  solution  of  an  ordinary 
differential  equation  of  the  nth  order  will  contain  n  arbitrary  constants. 
A  differential  equation  is  usually  said  to  be  solved  when  the  problem  is 
reduced  to  a  simple  quadrature,  that  is,  an  integration  of  the  form 
y  =  fftx^dx. 

Methods  of  Solving  Ordinary  Differential  Equations 

DIFFERENTIAL  EQUATIONS  OF  THE  FIRST  ORDER 

(1)  If  possible,  separate  the  variables;  that  is,  collect  all  the  x's  and  dx  on 
one  side,  and  all  the  y's  and  dy  on  the  other  side;  then  integrate  both  sides, 
and  add  the  constant  of  integration. 

(2)  If  the  equation  is  homogeneous  in  x  and  y,  the  value  of  dy/dx  in  terms 

of  x  and  y  will  be  of  the  form  —  =  /(-).     Substituting  y  =  xt  will  enable 

dx          \x] 

the  variables  to  be  separated.    Solution:  loge  x  =  f~fr\ H  C. 

(3)  The    expression    f(x,y)dx   +  F(x,y}dy     is     an     exact     differential     if 

—  =  — ^ — (  =  P,    say).     In    this    case  the    solution  of   f(x,y)dx  + 
oy  ox 

F(x,y)dy  =  0  is 

ff(x,y)dx  +  f[F(x,y)  -  fPdx]dy  =  C 

or  fF(x,y}dy  +  f\f(x,y)  -  fPdy]dx  =  C 

(4)  Linear  differential  equation  of  the  first  order:  —  -f-  f(x)-y  =  F(x). 

dx 

Solution:  y  =  e~p  {  ftpF(x)dx  +  c}  ,  where  P  =  ff(x)dx. 

(5)  Bernoulli's  equation:  - — |-  f(x)-y  =F(x)-yn.      Substituting    yl~n  =v 

ax 

dv 
gives  —  +  (1  —  ri)f(x)'V  =  (1  —  n)F(x),  which  is  linear  in  v  and  x. 

(6)  Clairaut's    equation:  y  =  xp  +  /(p),  where  p  =  dy/dx.     The  solution 
consists  of  the   family    of    lines    given   by  y  =  Cx  +  /(O ,  where  C  is  any 
constant,  together  with  the  curve  obtained  by  eliminating  p  between  the 
equations  y  =  xp  +  f(p)  and  x  +  f'(p)   =  0,  where  f'(p)  is  the  derivative  of 

DIFFERENTIAL  EQUATIONS  OF  THE  SECOND  ORDER 

(7)  -,—  ~   —n*y.     Solution:  y  =  Ci  sin  (nx  +€2) 
ax* 

or  y  =  Cz  sin  nx  +  Ci  cos  nx 


172  DIFFERENTIAL  AND  INTEGRAL  CALCULUS 

(8)  —  =  +n2y.     Solution:  y  =  Ci  sinh  (nx  +  C2) 

or  y  =  €3  enx  +  (74  e~nx 

(9)  --  =  /(y).     Solution:  x  =    f — - —  +  C2,  where  P  =  ff(y)  dy. 

VCi  +  2P 

(10)  ^  =  /(*).  Solution:  y  =  fPdx  +  Cix  +  C2,  where  P  =  f  j(x)dx 

or  y  =  xP  -  J*xf(x)dx  +  Cix  +  C2 

(11)  3—  =  f\-r )  •     Putting  —  =  z,  —  =  — ,  a;  =    /*• — -  -f  Ci  and  y  = 
ox2          \dxj  dx  dx2        dx  **  /(z) 

J  ~7T\  ~\~  Cz't  then  eliminate  z  from  these  two  equations. 

(12>  The  equation  for  damped  vibration:  ~-  +  26—  +  a2y  =  0. 

arc2  dx 

Case  I.  If  o2  -  62  >  0,  let  m  =  Va2  -  62.     Solution:  » 

y  =  Ci  e~bx  sin  (mx  +  ^2)  or  y  =    e~*x[Cz  sin  (mx)  +  C4  cos  (mx)] 
Case  II.     If  a2  -  62  =  0,  solution  is  y  =  e~bx[Ci  +  Ctx]. 

Case  III.  If  a2  -  62  <  0,  let  n  =  V&  -  a2.     Solution: 

y  =  Cie"**  sinh  (nx  +  C2)  or  y  =  C3e~(b+n)x  +  C4e~(6~n)a; 

d*y  dy  c 

(13)  3—  +  26- h  o2y  =  c.     Solution:  y  =  : 1-  yi,  where  y\  =  the  solu- 

ax2  ax  a2 

tion  of  the  corresponding  equation  with  second  member  zero  [see  (12)  above]. 

(14)  -^  +  26-^  +  a*y  =  c  sin(fec).     Solution: 
ax*  ax 

y  =  R  sin(kx  -  S)  +  yi,  where  R  =  c/V(a2  -  k2)2  +  462A;2, 

26fc 
tan  <S  =    2  _      ,  and  y\  —  the  solution  of  the  corresponding  equation  with 

second  member  zero  [see  (12)  above]. 

d2y  dy 

(15)  3^  +  2&3^  +  aZV  =  f(x)>     Solution:  y  =  y0  +  y\,     where     y0  =--  any 

particular  solution  of  the  given  equation,  and  y\  =  the  general  solution  of  the 
corresponding  equation  with  second  member  zero  [see  (12)  above]. 

If  62  >  a2,  y0  =  - 


b*  -  a2 

where  m,\  =  —  b  +  \/62  —  q2  and  mz  =  —  6  — 
If  62  <a2,  letm=Va2  -  62;  then^/o  = 

— e~*x  |  sin  (mx)J*ebx  cos  (mx)-/(x)da;  —  cos  (mx}febx  sin 
If  62  =  a2, 2/e  =  e~&a;  (  xj*eb*f(x)  dx  -  fx-J>xj(x)  dx  \  . 


GRAPHICAL  REPRESENTATION  OF  FUNCTIONS 

For  graphical  methods  in  statistics,  etc.,  see  W.   C.  Brinton's   "Graphical   Methods  for 

Presenting  Facts" 

EQUATIONS  INVOLVING  TWO  VARIABLES 

The  Curve  y  =  f(x).  To  represent  graphically  any  function,  y,  of 
a  single  variable,  x,  lay  off  the  values  of  x  as  abscissae  along  a  uni- 
formly graduated  horizontal  axis,  whose  positive  direc- 
tion (as  usually  chosen)  runs  to  the  right,  and  at  each 
point  on  this  z-axis  erect  a  perpendicular  (called  an  ordi- 
nate)  whose  length  represents  the  value  of  y  at  that 
point.  The  unit  of  measurement  for  the  y-sca\e,  whose 


positive. direction  (as  usually  chosen)  runs  upward,  need  jpIG    j 

not  be  the  same  as  the  unit  for  the  a>scale.     Draw  a 
smooth  curve  through  the  extremities  of  the  ordinates;  this  is  the  graph  of 
the  given  function  in  rectangular  co-ordinates,  or  the  curve  of  the  function. 

To  measure  graphically  the  rate  of  change  of  the  function  at  any  point  P 
(Fig.  1),  draw  the  tangent  atP;  then  rate  of  change  at  P  =  RT /PR,  where 
RT  and  PR  are  measured  in  units  of  the  y-axis  and  x-axis,  respectively. 
This  ratio,  which  is  positive  if  RT  runs  upward,  negative  if  RT  runs  down- 
ward, is  equal  to  the  derivative  of  the  function  at  the  point  P  (see  p.  157). 

Graphs  of  Important  Functions.  Figs.  2-9  show  the  graphs  (in  rec- 
tangular co-ordinates)  of  the  most  important  elementary  functions,  namely: 

The  linear  function,  y  =  mx  +  b  (Fig.  2). 

The  power  functions,  y  =  xn  [n  positive  (parabolic  type) ;  n  negative 
(hyperbolic  type)]  (Fig.  3). 

The  exponential  function,  y  =  10*  or  y  =ex,  and  the  logarithmic 
function,  y  =  Iog10  x  or  y  =  loge  x  (Fig.  4). 

The  trigonometric  functions  (Fig.  5),  and  the  inverse  trigonometric 
functions  (Fig.  6). 

The  hyperbolic  functions  (Figs.  7  and  8)  and  the  inverse  hyperbolic 
functions  (Fig.  9). 

Various  special  functions  (Figs.  10-12). 

By  a  slight  modification,  each  of  these  diagrams  may  be  made  to  represent 
a  somewhat  more  general  function  than  that  for  which  it  is  primarily  intended. 
For,  if  x  is  replaced  by  a;  —  a  in  the  equation,  this  merely  requires  re-number- 
ing the  x-axis  so  that  each  number  is  moved  a  units  to  the  left;  and  similarly, 
if  y  is  replaced  by  y  —  b  in  the  equation,  this  merely  requires  re-numbering 
the  y-axis  so  that  each  number  is  moved  b  units  downward.  (Such  a  change 
is  called  a  translation  of  the  curve  to  the  right,  or  upward.)  Further,  if  x  is 
replaced  by  x/c  [or  y  by  y/c]  in  the  equation,  it  is  merely  necessary  to  multiply 
each  of  the  numbers  written  along  the  x-axis  [or  y-axis]  by  c,  in  order  to 
adapt  the  graph  to  the  new  equation.  (Such  a  change  is  called  a  "stretch- 
ing" of  the  curve  along  one  of  the  axes.) 

Empirical  Curves.  Any  set  of  values  of  two  variables  x  and  y  can  be 
represented  by  plotting  the  points  (x,y)  on  rectangular  co-ordinate  paper,  and 
drawing  a  smooth  curve  through  these  points.  The  points  which  correspond 
to  actual  data  should  be  clearly  indicated  by  small  circles  or  crosses,  inter- 
mediate points  being  spoken  of  as  interpolated  points.  While  this  process  of 
graphically  interpolating  a  continuous  series  of  points  between  given  values  is 
usually  fairly  safe,  the  process  of  extrapolation — that  is,  extending  the  curve 
beyond  the  range  of  the  given  values,  is  dangerous. 

173 


174 


GRAPHICAL  REPRESENTATION  OF  FUNCTIONS 


Linear  function,  y  =  ma;  +  b. 
FIG.  2. 


/  o  i 

(Parabolic  Type)  (Hyperbo/fc  Type) 

Power  function,  y  •=  xn. 
FIG.  3. 


ff 


Exponential  function  (10*  or  ex~). 
Logarithmic  function  (logic  x  or  loge  x) . 

FIG.  4. 


Inverse  trigonometric  functions. 
FIG.  6. 


-4FJX 


Trigonometric  functions. 
FIG.  5. 

To  Find  a  Mathematical  Equation  to  Fit  a  Given  Empirical  Curve. 

This  problem  is  one  which  in  general  requires  much  patience  and  ingenuity. 
Only  the  simplest  cases  can  be  mentioned  here. 

CASE  1.  If  the  given  empirical  curve  is  a  straight  line,  then  the  law  con- 
necting the  given  values  of  x  and  y  is  y  =  mx  +  6,  where  ra  =  the  slope  of  the 
line,  and  6  =  the  value  of  y  at  the  point  where  the  line  crosses  the  2/-axis.  If 


EQUATIONS  INVOLVING  TWO  VARIABLES 


175 


i      z      3 


Hyperbolic  functions  and  inverse  hyperbolic  functions. 
FIG.  7.  FIG.  8.  FIG.  9. 


y.,<»» 


K-or...v 
jr.02-...s 

/r—aaJ 


FIG.  10. 


/f.-/.5-/I          /-f  **»•** 


FIG.  12. 


the  points  lie  only  approximately  on  a  straight  line,  the  best  position  for  this 
line  can  usually  be  found  by  stretching  a  black  thread  among  the  points;  or, 
assume  a  law  of  the  form  y  =  mx  -\-  b,  and,  by  substituting  in  this  formula  n 
pairs  of  values  of  x  and  y,  obtain  n  equations  connecting  the  coefficients 
ra  and  b;  various  pairs  of  these  equations  may  then  be  solved  for  m  and  6,  and 
the  average  of  the  results  taken.  Or,  if  great  accuracy  is  required,  all  nof  the 
equations  may  be  solved  for  m  and  6  by  the  method  of  least  squares  (p.  121). 

If  any  law  of  the  formj(x,y')  =  m-F(x,y)  +  6  is  suspected,  where  f(x,y)  and 
F(x,y)  are  any  expressions  involving  either  x  or  y  or  both  x  and  y,  such  a  law 
may  be  tested  by  plotting  F(x,y)  instead  of  x,  andf(x,y)  instead  of  y,  on  rec- 
tangular cross-section  paper,  and  seeing  whether  or  not  the  points  lie  on  a 
straight  line.  If  they  do,  the  form  of  the  law  is  verified,  and  the  values  of  m 
and  b  can  be  read  from  the  figure  as  before.  For  example,  if  j/2  =  mxy  +  &, 
a  straight  line  will  be  obtained  by  plotting  y2  against  xy.  Again,  if  xy  =  bx  + 
my,  a  straight  line  will  be  obtained  by  plotting  y  against  y/x,  since  the  equa- 
tion may  be  written  y  =  b  -f  m  (y/x)- 

CASE  2.  If  a  law  of  the  form  y  =  cxn  is  suspected,  plot  the  points  (x,y)  on 
logarithmic  paper  (see  below). 

CASE  3.  If  a  law  of  the  form  y  =c-10wa;[or  y  =  c-emx]  is  suspected,  plot 
the  points  (x,y)  on  semi-logarithmic  paper  (see  below). 


176 


GRAPHICAL  REPRESENTATION  OF  FUNCTIONS 


CASE  4.  If  the  given  curve  resembles  the  logarithmic  curve,  y  =  log  x, 
interchange  x  and  y  and  proceed  as  in  Case  3. 

CASE  5.  If  the  given  curve  is  a  wavy  line,  resembling  a  sine  or  cosine  curve, 
try  an  equation  of  the  form  y  =  a  sin  bx  or  y  =  a  cos  bx.  If  the  heights  of  the 
waves  diminish  as  x  increases,  try  an  equation  of  the  form  y  =  ae~nx  sin  bx. 
[NOTE.  Any  periodic  function  (satisfying  certain  simple  conditions)  can  be 
expressed  by  a  Fourier's  series  (p.  162)]. 

CASE  6.  A  great  variety  of  functions  can  be  represented  approximately  by 
a  polynomial  of  the  form  y  =  a  +  bx  +  ex2  -f  dx3  +  ex*  + .  .  . ,  the  first 
three  or  four  terms  being  usually  sufficient.  To  determine  the  coefficients 
a,  b,  c,  .  .  . ,  most  accurately,  substitute  in  the  formula  all  the  given  pairs  of 
values  of  x  and  y,  and  solve  the  resulting  equations  for  a,b,c, ...  by  the 
method  of  least  squares  (p.  121). 

CASE  7.  Many  simple  curves  can  be  represented  approximately  by  an 
equation  of  the  hyperbolic  form,  xy  =  c  -f  bx  -f-  ay,  where  a,  b,  and  c  are 
determined  by  substituting  the  co-ordinates  of  three  conspicuous  points  of  the 
curve.  The  lines  x  =  a  and  y  =  b  are  the  asymptotes  of  the  hyperbola. 
The  equation  may  also  be  written  (x  —  a)  (y  —  b)  =  k,  where  k  =  ab  +  c. 

Logarithmic  Cross-section  Paper.  In  this  form  of  cross-section  paper 
(Fig.  13),  the  distance  from  the  origin  to  any  point  on  the  x-  or  y-axis  is  equal 
to  the  logarithm  of  the  number  written  against  that  point.  Thus,  in  Fig.  13 
the  distances  (shown  for  clearness  on  two  auxiliary  scales  X  and  Y)  are  the 
logarithms  of  the  numbers  written  along  x  and  y. 
Y  v 


50      100  x 
"~~t   X 


FIG.  13. 


3          4         56789   10 

FIG.  14. 


Accurately  made  logarithmic  paper  can  be  obtained  from  the  principal 
dealers  in  draftmen's  supplies.  Logarithmic  paper  can  be  easily  con- 
structed, in  case  of  need,  by  copying  the  logarithmic  scale  from  any  ordinary 
slide  rule.  The  actual  figures  along  the  x-  and  y-axes  are  usually  left  for  the 
user  to  insert;  in  so  doing,  notice  that  the  numbers  .  .  .,0.01,  0.1,  1,  10, 
100,  .  .  . ,  or  such  of  them  as  may  be  needed  to  cover  any  given  range  of 
values,  must  be  placed  at  the  points  of  division  which  separate  the  main 
squares.  It  is  often  convenient,  however,  to  omit  the  decimal  point,  num- 


LOGARITHMIC  CROSS-SECTION  PAPER  177 

bering  each  square  independently  from  1  to  10.  The  length  of  the  side  of 
one  square  is  called  the  unit  or  base  of  the  logarithmic  paper;  the  larger  the 
unit,  the  finer  the  possible  subdivisions  of  the  scale. 

To  plot  a  point  (x,y)  on  logarithmic  paper,  for  example,  the  point  (3,5), 
means  to  find  the  point  of  intersection  of  the  vertical  line  marked  x  —  3  and 
the  horizontal  line  marked  y  =  5.  In  interpolating  between  two  lines, 
account  should  be  taken  of  the  fact  that  the  divisions  are  not  of  uniform  length. 

Any  equation  of  the  form  y  =  cxn  when  plotted  on  logarithmic  paper  will 
be  represented  by  a  straight  line  whose  slope  is  n.  For,  if  y\  =  cx\n  and 
yt  =  cx2n,  then  yi/yz  =  (xi/xz)11,  or  (log  y\  —  log  yz) /(log  x\  —  log  xz)  =  n. 
The  slope  must  be  measured  by  aid  of  an  auxiliary  uniform  scale. 

EXAMPLE.  Let  y  =  a;3'2.  When  x  =  1,  y  =  1;  plot  this  point  A  on  the  logarithmic 
paper,  and  draw  the  straight  line  AE  with  a  slope  equal  to  %  (Fig.  13).  By  the  aid  of 
this  line,  the  value  of  y  for  any  value  of  x  between  1  and  100  can  be  read  off  directly; 
for  example,  if  x  =  2.50,  y  =  3.95,  as  shown  by  dotted  lines,  so  that  (2.50)»/z  =  3.95. 
To  find  the  value  of  y  for  any  value  of  x  outside  this  range,  note  that  moving  the  decimal 
point  2  places  in  x  is  equivalent  to  moving  it  3  places  in  y.  The  line  shown  in  Fig.  13 
is  thus  equivalent  to  a  complete  table  of  three-halves  powers. 

It  will  be  noticed  that  this  line  crosses  four  squares  of  the  logarithmic  paper.  By 
superposing  these  four  squares  the  whole  diagram  may  be  condensed  into  a  single  square 
(Fig.  14),  in  which,  however,  the  scales  for  x  and  y  now  give  only  the  sequence  of  digits 
in  the  answer,  the  position  of  the  decimal  point  having  to  be  determined  by  inspection. 

To  determine  whether  a  given  set  of  values,  z  and  y,  satisfies  a  law 
of  the  form  y  =  cxn,  plot  the  values  on  logarithmic  paper,  and  see  whether 
they  lie  on  a  straight  line;  if  they  do,  then  the  given  values  satisfy  a  law  of 
this  form ;  moreover,  the  slope  of  the  line  gives  the  value  of  n,  and  the  value 
of  y  when  x  =  1  gives  the  value  of  c. 

If  the  plotted  points  fail  to  lie  exactly  in  line,  but  form  a  curve  slightly  concave  up- 
ward, try  subtracting  some  constant  b  from  all  the  y's,  that  is,  move  each  point  downward 
a  distance  equal  to  b  units  of  the  y-scale  at  that  point.  If  it  proves  possible  to  choose  b 
so  that  the  resulting  points  lie  in  line,  then  the  original  values  obey  a  law  of  the  form 
y  —  b  =  cxn,  where  n  is  again  the  slope  of  the  line,  and  c  is  the  value  of  y  —  b 
when  x  =  1.  (Conversely,  if  the  curve  is  concave  downward,  try  adding  b  to  all  the 
y's;  that  is,  move  each  point  upward; if  the  new  points  lie  in  line,  the  original  values 
obey  a  law  of  the  form  y  +  b  =  cxn.)  Another  method  of  "straightening"  the 
curve  consists  of  adding  some  constant,  ±  a,  to  all  the  values  of  x,  which  has  the 
effect  of  shifting  all  the  points  to  the  right  or  left  (by  varying  amounts) ;  if  this 
method  succeeds,  the  original  values  obey  a  law  of  the  form  y  =  c(x  +  a)n. 

Semi-logarithmic  Cross-section  Paper*.  This  form  of  paper  (Fig.  15) 
has  a  logarithmic  scale  along  y  and  a  uniform  scale  along  x.  The ''scale 
value,"  k,  of  the  paper  is  the  number  which  stands,  on  the  z-axis,  at  a  dis- 
tance from  the  origin  equal  to  the  width  of  one  of  the  main  horizontal  strips. 
Thus,  in  Fig.  15,  each  number  shown  along  the  auxiliary  scale  Y  is  the  loga- 
rithm of  the  corresponding  number  along  y,  and  each  number  shown  along 
the  auxiliary  scale  X  is  1/fcth  of  the  corresponding  number  along  x  (here 
k  =  5).  The  number  k,  which  may  be  chosen  at  pleasure,  should  be  taken 
equal  to  some  simple  integer,  as  1,  2,  or  5,  or  some  integral  power  of  10. 

In  preparing  the  paper  for  use  it  is  important  to  notice  that  the  numbers 
.  .  .,0.01,0.1,  1,  10,100,  .  .  .  {or  such  of  them  as  may  be  needed  in  any 
given  case)  must  be  placed  along  the  2/-axis  at  the  points  which  mark  the  main 
lines  of  division  between  the  horizontal  strips;  while  the  numbers  .  .  ., 
—  2fc,  —  k,  0,  +  k,  +  2k,  .  .  .  (or  such  of  them  as  may  be  needed)  must 
be  placed  along  the  rr-axis  at  uniform  intervals,  each  interval  (from  0  to  ft, 
from  A;  to  2k,  etc.)  being  equal  to  the  width  of  one  of  the  main  horizontal 
strips.  The  width  of  one  of  these  strips  is  called  the  unit  or  base  of  the  semi- 

*Made  by  the  Educational  Exhibition  Co..  26  Custom  House  St.,  Providence,  R.I. 
12 


178 


GRAPHICAL  REPRESENTATION  OF  FUNCTIONS 


10  x 

"zx 


logarithmic  paper;  the  larger  the  unit,  the  finer  the  possible  subdivisions  of 
the  scale. 

To  plot  a  point  (x,y},  as  a:  =  3,  y  —  5,  on  semi-logarithmic  paper  means  to 
find  the  point  of  intersection  of  the  vertical  line  marked  x  =  3  with  the 
horizontal  line  marked  y  =  5. 

Any  equation  of  the  form  y  = 
c-10mz  [or  y  —  c-emx]  when  plotted 
on  semi-logarithmic  paper  with  scale 
value  k,  will  be  represented  by  a 
straight  line  whose  slope  is  km  [or 
0.4343  few.].  By  a  suitable  choice  of 
the  scale  value  k,  any  given  range  of 
values  of  x  can  be  brought  within 
the  size  of  the  paper.  Note  that  e  = 
10°-4343. 

EXAMPLE.     Given  y  =  4-10-0-1*  [or  y  = 

4-e-0-1*].     In  Fig.  15,  when  x  =  0,  y  =  4.  _!  "  '  " ' 

By  plotting  this  point  (A)  on  the  semi- 
logarithmic  paper,  with  scale  value  5,  and  FIG.    15. 
drawing  through  it  a  straight  line  with 

slope  equal  to  —  0.5  [or  —  0.217]  a  graphical  representation  is  obtained  from  which,  for 
any  value  of  x,  the  corresponding  value  of  y  can  be  read  off.  If  it  is  desired  to  condense 
the  figure,  several  horizontal  strips  may  be  superposed  on  a  single  strip;  this  of  course 
renders  the  decimal  point  in  the  y-scale  undetermined  (unless  a  separate  y-scale  is 
provided  for  each  section  of  the  graph). 

In  order  to  determine  whether  a  given  set  of  values  of  x  and  y  satisfy 
a  law  of  the  form  y  =  c-Wmx  [or  y  =  c-emz],  plot  the  values  of  x  and  y  on 
semi-logarithmic  paper,  with  a  suitable  scale  value  k,  and  see  whether  they 
lie  on  a  straight  line;  if  they  do  so,  the  law  is  satisfied,  and  the  values  of  m 
and  c  may  be  found  as  follows:  m  =  the  slope  of  the  line  divided  by  k  [or  the 
slope  of  the  line  divided  by  0.4343/c],  and  c  =  the  value  of  y  when  x  =  0. 

If  the  plotted  points  fail  to  lie  exactly  in  line,  but  form  a  curve  slightly  concave  up- 
ward, try  subtracting  some  constant  b  from  all  the  y's,  and  plot  the  values  thus  modified; 
if  &  can  be  so  chosen  that  the  revised  points  lie  in  line,  then  the 
original  values  obey  a  law  of  the  form  y  —  b  =  c-10mx  [or  y  —  b  = 
c-emx],  where  m  and  c  are  to  be  found  as  before.  If  the  curve  is  con- 
cave downward,  add  b,  instead  of  subtracting;  and  replace  y  —  bby 
y  +  b  in  the  law. 

Curves  in  Polar  Co-ordinates.     Any  function,  r,  of  a  single  vari- 
able, f>,  can  be  represented  by  a  curve  in  polar  co-ordinates  (p.  137). 
Lay  off  the  given  values  of  6  as  angles,  the  initial  line  Ox  running 
toward  the  right,  and  the  counterclockwise  direction  about  the  origin          flQ     16 
being  taken  as  positive.     Along  the  terminal  side  of  each  angle  0, 
lay  off  the  corresponding  value  of  r,  forward  if  r  is  positive,  backward  if  r  is  negative; 
and  pass  a  smooth  curve  through  the  points  thus  determined. 

The  rate  of  change  of  r  with  respect  to  6  at  a  given  point  P  is  represented  graphically 
as  follows  (Fig.  16):  On  the  tangent  at  P  drop  a  perpendicular  OM  from  the  origin; 
then  r(MP/OM)  represents  the  rate  of  change,  dr/de,  provided  0  is  measured  in 
radians.  Specially  ruled  polar  co-ordinate  paper  is  supplied  by  dealers  in  drafting 
supplies. 

EQUATIONS  INVOLVING  THREE  VARIABLES 

The  Surface  z  =  f(x,  y).  Any  function,  z,  of  two  variables,  x  and  y, 
may  be  represented  by  a  surface,  as  follows:  Plot  the  given  pairs  of  values 
of  x  and  y  as  points  in  a  horizontal  x,  y  plane,  called  the  base  plane;  at  each 
of  these  points  erect  an  ordinate,  parallel  to  a  vertical  axis  z,  and  representing 


EQUATIONS  INVOLVING  THREE  VARIABLES 


179 


by  its  length  the  value  of  z  at  that  point.  Then  conceive  a  smooth  surface 
passed  through  the  extremities  of  these  ordinates:  this  surface  is  said  to  repre- 
sent the  function.  In  practice,  the  ordinates  may  be  made  by  implanting 
stiff  vertical  rods  in  a  horizontal  board  of  soft  wood  which  serves  as  the  base 
plane;  the  surface  may  then  be  constructed  by  filling  in  the  spaces  with  plaster 
of  Paris.  Or,  more  simply,  pieces  of  cardboard  may  be  cut  out  to  represent 
parallel  plane  sections  of  the  surface,  and  then  stood  on  edge  in  slots  cut  in 
the  board  to  receive  them.  The  units  employed  along  x,  y,  and  z  need  not  be 
equal  to  each  other. 

Contour-line  Charts.  All  the  points  of  a  surface  z  =  f(x,  y)  which  are 
at  any  given  height  above  the  base  plane  form  a  curve  on  the  surface,  called 
a  contour  line  of  the  surface.  If  each  of  these  contour  lines  be  projected 
on  the  base  plane,  and  each  labeled  with  the  value  of  z  to  which  it  corresponds, 
a  complete  representation  of  the  function  z  =  f(x,  y)  is  obtained,  all  in  one 
plane.  A  topographical  map,  with  contour  lines  showing  elevations  above 
the  sea,  and  a  weather  map,  with  contour  lines  showing  barometric  pressure, 
are  familiar' examples.  If  there  are  several  values  of  z  corresponding  to  any 
given  point  (x,  y},  there  will  be  several  contour  lines  whose  projections  pass 
through  that  point. 

Contour-line  Charts  for  Simultaneous  Equations  [of  the  form  z  - 
f(x,y),  w  =  F(x,y~)].  In  Fig.  17,  plot  the  function  z  =  f(xty)  by  contour 
lines  on  an  x,y  plane,  and  plot  the  function  w  =  F(x,y) 
by  contour  lines  on  the  same  x,y  plane.  Then  every 
point  on  the  diagram  (either  directly  or  by  interpola- 
tion) is  the  intersection  of  four  curves — an  z-curve, 
a  y-curve,  a  z-curve,  and  a  w-curve.  Here,  by 
"curve"  is  meant  any  line,  straight  or  curved.  By 
the  aid  of  such  a  diagram,  when  the  values  of  any 
two  of  these  four  variables  are  given,  the  values  of 
the  other  two  can  be  found.  The  method  of  use 
consists  simply  in  entering  the  diagram  along  the  two 
given  curves  (or  lines),  tracing  them  to  their  point  of 
intersection,  and  then  coming  out  again  along  the 
two  curves  (or  lines)  whose  values  are  required.  The  best  manner  of  num- 
bering the  curves  is  indicated  in  the  figure. 

Alignment  Charts  for  Three  Variables,  t,  u,  v.  Any  relation  between 
three  variables,  t,  u,  v,  which  can  be  thrown  into  one  of  the  forms  listed  in 
later  paragraphs,  can  be  represented  graphically  by  a  very  convenient  form 
of  diagram  called  an  alignment  chart.  In  the  simplest  form  of  an  alignment 
chart  for  three  variables  there  are  three  scales  (straight  or  curved),  along 
which  the  values  of  the  three  variables,  t,  u,  v,  are  marked  in  such  a  way  that 
any  three  values  of  t,  u,  v  which  satisfy  the  given  equation  are  represented 
by  three  points  which  lie  in  line.  Hence,  if  the  values  of  any  two  of  the  vari- 
ables are  given,  the  corresponding  value  of  the  third  can  be  found  by  simply 
drawing  a  straight  line  through  the  two  given  points  and  reading  the  value 
of  the  point  where  it  crosses  the  third  scale. 

The  most  important  methods  of  constructing  alignment  charts  for  three 
variables  are  described  below.  Where  several  methods  are  applicable  in  a  given 
case,  the  best  one  must  be  determined  largely  by  trial.  For  further  informa- 
tion see  M.  d'Ocagne,  "Traite  de  Nomographie"  (Gauthier-Villars,  Paris); 
Carl  Runge,  "Graphical  Methods"  (Columbia  University  Press) ;  J.  B.  Peddle, 
"Construction  of  Graphical  Charts"  (McGraw-Hill);  see  also  page  185. 


180 


GRAPHICAL  REPRESENTATION  OF  FUNCTIONS 


Notation.  In  each  of  the  equations  which  follow,  U  stands  for  any 
function  of  u  alone,  V  for  any  function  of  v  alone,  and.Fi(0,  Fz(t)  for  any  func- 
tions of  t  alone.  Any  of  these  functions  may  reduce  to  a  constant.  The 
axes  of  x,  y,  and  y'  which  are  mentioned  are  of  merely  temporary  use  in  con- 
structing the  diagram,  and  the  letters  x,  y,  y'  should  not  be  written  on  the 
chart.  It  is  not  necessary  that  the  axes  be  at  right  angles,  provided  the  x 
of  a  point  is  always  measured  parallel  to  the  z-axis,  and  its  y  parallel  to  the 
y-axis. 

Method  1.     Given,  an  equation  which  can  be  thrown  into  the  form 

U-Fi(£)  +  V-Fi(t)   =  1, 

where,  for  the  given  range  of  values  of  u  and  v,  the  largest  variations  in  U 
and  V  are  less  than  a  certain *number  m. 

Draw  a  pair  of  (temporary)  x,y  axes  (Fig.  18),  and  through 
the  point  x  =  1  draw  a  third  axis,  which  may  be  called  the  axis 
of  y',  parallel  to  the  axis  of  y.  In  ordinary  cases,  the  unit  of 
measurement  along  x  should  be  nearly  equal  to  the  full  width 
of  the  paper.  Now  choose  a  unit  for  y  and  y'  such  that  m 
times  this  unit  will  about  equal  the  height  of  the  paper,  and 
plot,  in  the  usual  way,  the  points  (x,y)  given  by 


FIG.   18. 


labeling  each  point  with  the  value  of  t  to  which  it  corresponds.  Connect 
these  points  by  a  smooth  curve,  which  gives  the  £-scale  of  the  diagram.  [If 
Fi(t)/Fz(t)  =  a  constant,  the  £-scale  will  prove  to  be  a  straight  line  parallel 
to  the  j/-axis.] 

Then,  using  the  same  units  as  above,  plot  along  y  the  points  given  by 
y  =  U,  labeling  each  point  with  the  corresponding  value  of  u\  and  plot  along 
yf  the  points  given  by  y'  =  V,  labeling  each  of  these  points  with  the  corre- 
sponding value  of  v.  This  gives  the  u-  and  v-scales  of  the  diagram.  The 
three  scales  being  thus  constructed,  the  a;-axis  may  now  be  erased,  and  the 
diagram  is  ready  for  use.  Any  three  points  t  ,  u,  v  which  lie  in  line  correspond 
to  three  values  of  t,  u,  v,  which  satisfy  the  given  equation.  The  numbering 
on  each  scale  should  be  shown  at  sufficiently  frequent  intervals  to  permit  of 
easy  interpolation. 


200 
100 

'  5 

EO       4 
10 
5 
2 


U'V  '     "  f 

FIG.  19. 


u 

200  - 


-100- 


O 
50  J 


EXAMPLE  1  (Fig.  19).  Let  u»1>tt  =  t.  By  taking  the  logarithm  of  both  sides,  and 
dividing  through  by  log  t,  reduce  the  equation  to  the  form  (log  u)  (I/log  0  +  (log  t>)  X 
(1.41/log  0  -  1.  Here  U  =  log  u,  V  =  log  v,  Fi(t)  <*>  I/log  tt  F*(t)  -  1.41/log  t,  and 
Z  -  1.41/2.41  -  0.585,  y  -  (l/2.41)log  t. 


ALIGNMENT  CHARTS  181 

EXAMPLE  2  (Fig.  20).  Let  v  =  ut  +  16«2,  which  reduces  to  the  form  (-  u/16)(l/0 
+  (»/16)(l//2)  =  1.  Here  U  --  u/16,  V  =  t>/16,  Fi(t)  =  1/t,  F2«)  •=  1/«J  and 
x  -  1/(1  +  0,  2/  =  «V(1  +  0- 

NOTE.  If  m  =  oo  ,  values  of  «  and  v  which  give  large  values  of  U  and  7  cannot  be 
shown  within  the  limits  of  the  paper.  In  such  cases,  the  chart  may  be  supplemented 
by  a  second  chart,  made  according  to  Method  2,  below. 

Method  2.     Given,  an  equation  which  can  be  thrown  into  the  form 


,  _  -, 


U  V 

where,  for  the  given  range  of  values  of  u  and  v,  the  largest  variation  in  U  is 
less  than  a  certain  number  m.  and  the  largest  variation  in  V  is  less  than  a 
certain  number  n. 

Draw.a  pair  of  temporary  x,y  axes,  and  having  chosen  a  unit  for  the  z-axis 
equal  to  about  (l/m)th  of  the  width  of  the  paper,  and  a  unit  for  the  f/-axis 
equal  to  about  (l/n)th  of  the  height,  plot  the  points  (x,y)  given  by 

x  =  Fi(t),  y  =F2(0, 

labeling  each  point  of  this  curve  with  the  value  of  t  to 
which  it  corresponds.  Connect  these  points  by  a  smooth 
curve,  which  gives  the  <-scale  of  the  diagram.  [If 
FiCO/ACO  =  a  constant,  the  <-scale  will  be  a  straight  line 
through  the  origin.] 

Then,  using  the  same  units  as  above,  plot  along  x  the 
values  of  £7,  labeling  each  point  with  the  corresponding 
value  of  u',  and  plot  along  y  the  values  of  F,  labeling 
each  point  with  the  corresponding  value  of  v.  This  gives 
the  u-  and  v-scales  of  the  diagram.  On  the  chart  as 
thus  completed,  any  three  points  t,  u,  v  which  lie  in  line 
correspond  to  three  values  of  t,  u,  v  which  satisfy  the 
given  equation. 

EXAMPLE  (Fig.  21).  Let  t  =>  (uv)/(u  +  »),  which  may  be  written  in  the  form 
t/u  +  t/v  =  1.  Here  U  =  u,  V  =  v,  Fi(t)  =  t,  F2(t)  -  t. 

NOTE.  If  TO  =  oo  and  n  =  oo  ,  values  of  u  and  v  which  give  large  values  of  U  and 
V  cannot  be  shown  within  the  limits  of  the  paper.  In  such  cases  the  chart  may  be 
supplemented  by  a  second  chart,  made  according  to  Method  1,  above. 

Method  3.  Given,  an  equation  which  can  conveniently  be  thrown  into 
the  form 

F2(0  =  V'Fi(t)  +  U, 

where,  for  the  given  range  of  values  of  t,  the  largest  variation  in  Fi(C)  is  less 
than  a  certain  number  m,  and  the  largest  variation  inFz(f)  is  less  than  a  certain 
number  n. 

Draw  a  pair  of  temporary  x,y  axes,  and,  having  chosen  a  unit  for  x  equal 
to  about  (l/m)th  of  the  width  of  the  paper  and  a  unit  for  y  equal  to  about 
(l/n)th  of.  the  height,  plot  the  points  (x,y)  given  by 

*  =  Fi(£),  y  =F«(0, 

labeling  each  point  of  the  curve  with  the  value  of  I  to  which  it  corresponds. 
Connect  these  points  by  a  smooth  curve,  which  forms  the  <-scale.  Next, 
using  the  same  unit  for  y  as  above,  plot  along  the  y-axis  the  values  of  U, 
labeling  each  point  with  the  corresponding  value  of  u.  This  gives  the  w-scale. 
Finally,  with  the  origin  as  center,  and  any  convenient  radius,  draw  a  circle 
cutting  the  z-axis  in  A.  Along  this  circular  arc,  starting  from  A  in  the  coun- 
terclockwise direction,  lay  off  the  angles  whose  slopes  are  equal  to  F, 
labeling  each  point  of  the  arc  with  the  value  of  v  to  which  it  corresponds. 


182  GRAPHICAL  REPRESENTATION  OF  FUNCTIONS 

This  gives  the  v-scale,  which  in  this  case,  however,  plays  a  peculiar  role,  since, 
in  using  this  form  of  chart,  two  straight  lines  are  required  instead  of  one. 
Thus: 

In  order  to  determine  whether  three  values,  t,  u,  v, 
satisfy  the  given  equation,  lay  one  straight  line  through 
the  points  t  and  u,  and  another  straight  line  through 
the  point  v  and  the  origin;  if  these  lines  are  parallel, 
the  three  values  of  t,  u,  v  satisfy  the  equation.  It 
will  be  noticed  that  the  function  of  the  r-scale  here  is 
to  measure,  in  a  certain  sense,  the  slope  of  the  line 
joining  t  and  u.  A  chart  of  this  type  may  be  .called 
"  an  alignment  chart  with  a  sliding  scale  for  one  of  the 
variables."  , 

EXAMPLE  (Fig.  22).  Let  sin  u  =  sin  60°  sin  t  —  cos  60°cos  t 
cos  v,  which  may  be  put  in  the  form 

(sin  60°  sin  t)  =  cos  v  (cos  60°  cos  t)  +  sin  u. 
Here  Fi(0  =  cos  60°  cos  t,  Fz  (0  =  sin  60°  sin  t,  U  =  sin  u,  V  =  cos  v. 

Method  4.     Given,  an  equation  which  can  be  reduced  to  the 
form 

C7-F(0  +  V  =  0, 


sin  u- fin  60°iin  t-  cos  60cost  cos  v 

FIG.  22. 


where,  for  the  given  range  of  values  of  u  and  v,  the  largest  varia- 
tions  in  U  and  V  are  less  than  a  certain  number  m. 

In  Fig.  23,  draw  temporary  axes  x,  y,  and  y'  ,  and 
choose  the  units  as  in  Method  1.  To  construct  the 
f-scale,  which  will  now  coincide  with  the  a;-axis,  plot 
along  x  the  points  for  which 

1 

~  1  +  F(t)  ' 

labeling  each  point  with  the  value  of  t  to  which  it  cor- 
responds. The  it-scale,  along  the  axis  of  y,  and  v- 
scale,  along  the  axis  of  y'  ,  are  constructed  exactly  as 
in  Method  1,  and  the  finished  chart  is  used  in  the 
same  way. 


FIG.  23. 


15000 


V 

150000 


10000 
0 


FIG.  24. 


EXAMPLE  (Fig.  24).  Let  v  —  0.196  thi,  where  u  is  to  range  from  0  to  15,000  and  v 
from  0  to  150,000.  The  equation  may  be  written  in  the  form  (-  10  u)  (0.0196**)  +  v 
=  0.  Here  U  =  -  10  u,  V  =  v,  F(t)  =  0.0196<«. 

NOTE.  If  m  =  co  ,  values  of  u  and  v  which  give  large  values  of  U  and  V  cannot  be 
shown  within  the  limits  of  the  paper. 


EQUATIONS  INVOLVING  FOUR  VARIABLES 

[For  simultaneous  equations  of  the  form  z  =  f(x,y),  w  =  F(x,y),  see  p.  179.] 
Alignment  Charts  for  Four  Variables.     The  extension  of  the  methods 
of  the  alignment  chart  to  the  casex  of  four  variables,  say  r,  s,  u,  v,  consists 
essentially  in  replacing  the  f-scale  of  the  earlier  diagram  by  a  network  of  two 
scales,  one  for  r  and  one  for  s.     The  point  where  a  curve  r  =  r\  and  a  curve 
s  =  si  intersect  may  be  spoken  of  as  the  point  (ri,si).     In  the  following  equa- 
tions, U  denotes  as  before  any  function  of  u  alone,  V  any  function  of  v  alone; 
while  Fi(r,s)  and  Fz(r,s)  represent  any  functions  of  r  and  s. 
Method  la.     Given,  an  equation  of  the  form 

E7-Fi(r,«)  +  V-F2(r,s)   =  1. 


EQUATIONS  INVOLVING  FOUR  VARIABLES 


183 


Draw  axes  x,  y,  and  y'  as  in  Method  1,  and  plot  the  network  of  curves  given 
by  the  equations 


[To  do  this  (Fig.  25),  find  the  point  (x,y)  that  corresponds  to  each  given  pair 
of  values  of  r  and  s,  by  direct  substitution  in  the  equations  for  x  and  y.  Con- 
nect all  the  points  for  which  r  =  1  by  a  curve,  and  label  it  r  =  1;  connect 
all  the  points  for  which  r  =  2  by  another  curve,  and  label  it  r  =  2  ;  etc.  This 
gives  the  family  of  r-curves.  Similarly,  through  all  the  points  for  which 
a  =  1  draw  a  curve  labeled  s  =  1;  through  all  the  points  for  which  s  =  2 
draw  a  curve  labeled  s  =  2;  etc.  This  gives  the  family  of  s-curves,  intersect- 
ing the  family  of  r-curves.  Note,  however,  that  if  it  is  possible  to  eliminate 
s  (or  r-)  from  the  equations  that  give  x  and  y,  the  resulting  equation  in  x,  y, 
and  r  (or  x,  y,  and  s)  can  often  be  plotted  directly  for  each  given  value  of  r 
(or  of  s).] 

Next,  construct  the  u-  and  v-scales  along  the  axes  of  y  and  y'  as  in  Method  1. 
[The  letters  x,  y,  and  y',  and  the  units  used  in  plotting  along  these  axes,  should 
be  omitted  from  the  finished  diagram,  as  should  also  the  axis  of  x.] 

In  the  chart,  as  thus  completed,  any  three  points,  (r,s),  u,  and  v  which  lie 
in  a  straight  line,  correspond  to  values  of  r,  s,  u,  v  which  satisfy  the  given 
equation.  Hence,  when  any  three  of  these  four  values  are  given,  the  fourth 
can  be  found  from  the  chart. 


FIG.  25. 


O"!  2    3       i          5 

FIG.  26. 


Method  2a.     Given,  an  equation  of  the  form 


+ 


=  1. 


U  V 

Draw  axes  of  x  and  y  as  in  Method  2,  and  plot  the  network  of  curves  given  by 
x  =Fi(r,s),  y  =Fa(r,s). 

To  do  this,  follow  the  plan  outlined  for  a  similar  case  under  Method  la. 
labeling  each  curve  pf  the  r-family  (Fig.  26)  with  the  corresponding  value  of  r. 
and  each  curve  of  the  s-family  with  the  corresponding  value  of  s.  Next, 
construct  the  u-  and  v-scales  along  the  x-  and  y-axes,  precisely  as  in  Method  2. 
Then  any  three  points,  (r,s),  u,  and  v,  which  lie  in  a  straight  line  correspond 
to  values  of  r,  s,  u,  v  which  satisfy  the  given  equation. 

Method  3a.     Given,  an  equation  of  the  form 
F2(r,s)   =  F-Fx(r,s)  +  U. 

Draw  axes  of  x  and  y,  as  in  Method  3,  and  plot  the  network  of  curves  given 
by  x  =  Fi(r,s),  y  =  F2(r,s),  following  the  plan  outlined  for  a  similar  case 
under  Method  la,  and  labeling  each  curve  of  the  r-family  (or  s-family)  with 
the  valvte  of  r  (or  s)  to  which  it  corresponds.  Next,  construct  the  w-scale 


184 


GRAPHICAL  REPRESENTATION  OF  FUNCTIONS 


along  the  y-axis,  and  the  v-scale  along  a  circular  arc,  precisely  as  in  Method  3. 
Then  any  three  points,  (r,s)  u,  and  v,  which  are  so  related  that  the  line 
through  (r,s)  and  u  is  parallel  to  the  line  joining  v  with  the  origin,  will  corre- 
spond to  values  of  r,  s,  u,  v  which  satisfy  the  given  equation. 

EXAMPLE  for  Method  3a  (Fig.  27).     Let  cot  v  =  cot  r  cos  s  +  esc  r  sin  s  cot  u,  which 
may  be  written  (cos  r  cot  s)  =  cot  v  (sin  r  esc  s)  —  cot  u.     Here  U  =  —  cot  u,  V  =  cot  v, 

xz  y2  x2  w2 

Fi(r,s)  =  sin  r  esc  s,  Fz(r,s)  =  cos  r  cots,  whence  — —  +  • —  =1.    .    ,     —   —     =  1, 

csc2s       cot2s  sm2r         cos2r 

BO  that  the  s-curves  are  ellipses  and  the  r-curves  hyperbolas. 

Parallel  Charts,  or  Proportional  Charts,  for  Four  Variables.     In  the 

following  methods  of  representation  there  are  four  scales,  one  for  each  of  the 
four  variables,   and  the  method   of  using  the 
diagram   consists   in   connecting  two  pairs  of 
points  by  parallel  lines. 

Method  A.     Given,  an  equation  of  the  form 

R  -  S  =  U  -  V 

where  R,  S,  U,  V  are  any  functions  of  the 
variables  r,  s,  u,  v,  respectively.  [It  will  be 
noted  that  any  proportion  R/S  =  U/V  can  at 
once  be  thrown  into  this  form  by  taking  the 
logarithm  of  both  sides.] 

In  Fig.  28,  draw  four  vertical  axes,  yi,  yz,  y'l, 
y'z,  such  that  the  distance  between  y\  and  2/1 
(which  may  be  zero)  is  equal  to  the  distance 
beween  yz  and  3/2,  and  so  that  the  four  zero 
points  lie  in  line.  Along  these  axes,  using  the 
same  unit  for  all,  plot  the  points  given  by  yi  =R, 
y'l  =  S,  yz  =  U,  2/2  =  V,  and  label  each  point 
with  the  value  of  r,  s,  u,  or  v  to  which  it  cor- 
responds. (The  letters  y\,  yz,  y'l,  y'z  are  tem- 
porary, and  should  not  appear  on  the  diagram.) 
Then  if  the  line  joining  two  points  r  and  u  is 
parallel  to  the  line  joining  two  points  s  and  v, 
the  four  values  of  r,  s,  u,  v  will  satisfy  the  given 
equation.  In  this  and  the  following  methods, 
a  parallel  ruler,  or  a  pair  of  draf  tman's  triangles, 

will  be  useful  in  reading  the  chart.  A  "key"  stating  which  points  are  to 
be  joined  with  which,  should  be  clearly  given  on  the  diagram. 

EXAMPLE  (Fig.  28).  Let  32.2  vr  =  us2,  or  log  r  -  2  log  s  =  log  u  -  log  (32.2  »). 
Here  R  =  log  r,  S  =  2  log  s,  U  =  log  u,  V  =  log  (32.2  r). 

Method  B.     Given,  an  equation  of  the  form 

R,  _  J7 
S  ~    V 

In  Fig.  29,  draw  a  pair  of  axes,  x,y,  and  parallel  to  them  (or  coinciding 
with  them)  a  second  pair  of  axes,  xi,yi.  Using  any  convenient  horizontal 
unit,  plot  along  x  and  xi  the  points  given  by  x  =  R,  x\  =  U,  and  using  any 
convenient  vertical  unit,  plot  along  y  and  y\  the  points  given  by  y  =  S,  y\  =  V. 
Label  each  point  with  the  value  of  r,  s,  u,  v,  to  which  it  corresponds.  (The 
letters  x,  y,  x\,  y\  should  not  appear  on  the  diagram.)  Then  if  the  line  joining 
two  points  r  and  s  is  parallel  to  the  line  joining  two  points  u  and  v,  the  four 
values  r,  s,  u,  v  will  satisfy  the  given  eouation. 


COt  y  -  cot  r  cos  s  +  esc  r  sin  s  cot  a 
Vr,  Connect^?* %%»}  by  Parallel  Una. 

FIG.  27. 


EQUATIONS  INVOLVING  FOUR   VARIABLES 


185 


Method  C.     Given,  an  equation  of  the  form 

V 
U 


R  -  S 


In  Fig.  30,  take  a  pair  of  axes,  x,y,  and  through  the  point  x  =  1  draw  a 
third  axis,  y',  parallel  to  y.  Also,  take  a  second  pair  of  axes,  £2,2/2,  parallel 
to  (or  coinciding  with)  the  axes  of  x  and  y.  Having  chosen  a  suitable  unit 
for  x  and  £2,  and  a  suitable  unit  for  y,  y',  and  3/2,  lay  off  the  values  of  R  and 


500- 
100 


M  /'K? 

r  s         K'Y 

40 

<y 
•20 


•  US  * 

FIG.  28. 


v      5          Y      *y 

100- 

* 

-10. 
-5.     6- 

5- 

4- 

by  Para 

1  y  and  v\ 
lei  Lines. 

50- 

rU>      5 

3- 

:  x\. 

5 

~'5     4 

3 

•    \ 

NV    I             1 

V54 

5     6 

1    ' 

'    .        i           .              .                 ^V* 

i  r* 

i 

.             1  I  3    4         5 

6 

U  (X,) 


FIG.  29. 


<S  along  y  and  yf,  respectively,  labeling  each  point  with  the  value  of  r  or  8  to 
which  it  corresponds;  and  lay  off  the  values  of  U  and  V  along  xt  and  j/2,  label- 
ing each  point  with  the  value  of  u  or  v  to  which  it  corresponds.  Then  if  the 
line  joining  two  points  r  and  s  is  parallel  to  the  line  joining  two  points  u  and  v, 
the  four  values  r,  s,  u,  v  will  satisfy  the  given  equation.  This  form  of  chart 
is  sometimes  called  a  "Z-chart." 

For  further  examples,  see  R.  C.  Strachan,  "  Nomographic  Solutions  for 
Formulas  of  Various  Types,"  Trans.  Am.  Soc.  Civil  Engineers,  vol.  78,  1915. 


VECTOR  ANALYSIS 

Many  problems  involving  directed  magnitudes  can  be  advantageously 
treated  by  the  methods  of  vector  analysis.  The  following  is  a  brief  sum- 
mary of  the  principal  definitions  and  formulae. 

A  set  of  arrows,  each  arrow  having  a  given  length  and  pointing  in  a  given 
direction,  is  called  a  set  of  vectors,  provided  they  combine  by  addition  ac- 
cording to  the  parallelogram  law  (see  below).  Notation:  a  or  a  for  a  vector; 
a  or  |  a  |  for  its  length.  Two  "  free"  vectors  are  equal  if  they  have  the  same 
length  and  point  in  the  same  direction;  two  "sliding"  vectors  are  equal  if 
they  have  the  same  length  and  direction,  and  also  lie  in  the  same  line. 

A  scalar  is  any  real  number,  positive,  negative,  or  zero. 

Addition  of  vectors. — If  an  arrow  a  is  immediately  followed,  tip  to  tail,  by 
a  second  arrow  b,  then  the  arrow  which  runs  from  the  beginning  of  a  to  the  end 
of  b  is  called  the  sum  of  a  and  b,  denoted  by  a  +  b.  Conversely,  if  a  +  x  = 
b,  then  x  =  b  —  a.  The  laws  of  operation  for  +  and  —  are  the  same  as  in 
ordinary  algebra  (pp.  112,  124).  If  m  is  a  scalar,  then  ma  means  a  vector 
having  the  same  direction  as  a,  and  m  times  its  length. 


186  VECTOR  ANALYSIS 

Multiplication  of  vectors  is  of  two  kinds,  as  follows: 

The  scalar  product,  or  dot  product,  of  two  vectors  a  and  b,  denoted  by 
a-b  —  or  sometmes  by  Sab,  or  by  (ab)  in  round  parentheses  —  is  defined  as  the 
scalar  quantity  ab  eos  0,  where  0  is  the  angle  between  a  and  b. 

EXAMPLE.  If  P  is  a  force  whose  point  of  application  moves  along  a  vector  distance  x, 
then  F'X  =  work  done  by  F  during  this  displacement. 

Peculiarities  of  scalar  products:  (1)  Since  a;b  is  not  a  vector,  expressions 
like  (a-b)  -C.  will  not  occur;  (2)  from  a-x  =  a-y  we  cannot  infer  that  x  =  y, 
hence,  quotients  will  not  occur;  (3)  from  a-b  =  0,  it  follows  that  a  is  per- 
pendicular to  b  (unless  a  or  b  is  zero)  . 

On  the  other  hand,  scalar  products  are  like  ordinary  products  in  the  follow- 
ing respects:  a-b  =  b-a,  and  (a  +  b)-(c  +  d)  =  a-c  +  a-d  +  b-c  -f-  b-d; 
also,  m(a-b)  =  (ma-b)  =  a*(mb),  where  m  is  any  scalar. 

The  vector  product,  or  cross  product,  of  two  vectors  a  and  b,  denoted  by 
axb  —  or  sometimes  by  Vab,  or  by  [ab]  in  square  brackets  —  is  defined  as  the 
vector  whose  length  is  ab  sin  6,  where  0  is  the  angle  between  a  and  b,  and  whose 
direction  is  perpendicular  to  the  plane  of  a  and  b  (in  such  a  sense  that  a  right- 
handed  screw  advancing  along  axb  would  turn  a  toward  b). 

EXAMPLE.  If  F  is  a  force  acting  on  a  particle  whose  radius  vector  is  r,  then  rxF 
=  the  torque  of  F  about  the  origin. 

Peculiarities  of  vector  products:  (1)  axb  =  —  bxa,  so  that  the  order  of  the 
factors  is  always  important;  (2)  axa  =  0;  (3)  it  is  not  true  that  ax(bxc)  = 
(axb)xc;  (4)  from  axx  =  axy  it  does  not  follow  that  x  =  y;  hence,  quo- 
tients will  not  occur;  (5)  from  axb  =  0,  it  follows  that  a  and  b  are  parallel 
(unless  a  or  b  is  zero). 

On  the  other  hand,  as  in  ordinary  algebra 

(a  +  b)x(c  +  d)  =  axe  +  axd  +  bxc  +  bxd, 
provided  the  order  of  factors  in  each  product  is  preserved;  also, 
m(axb)   =  (ma)xb  =   ax(mb),  where  m  is  any  scalar.     Further  laws  are: 

a-(bxc)  =  b-(cxa)  =  c-(axb);   and  ax  (bxc)  =  (a-c)b  —  (a-b)c. 
Vector  Differentiation.     If  r  =  f  (f)  gives  a  vector  r  as  a  function  of  a 
scalar  t,  then  dr/dt  =  lim}[f(i  +  AO   —  f(0]/A*}  as  At  approaches  zero. 

d(a  +  b)  =  d&  +  db,     d(ma)  =  m(da)  +  (dm)  a, 
d(a-b)  =  (da)-b  +  a-(db),     d(axb)  =  (da)xb  + 


EXAMPLE.  If  r  =  f  (<)  gives  the  position-vector  of  a  moving  particle  as  a  function  of 
the  time  t,  then  dr/dt  =  its  vector  velocity,  v,  and  dv/dt  =  its  vector  acceleration,  a. 
If  m  and  n  are  unit  vectors  in  the  direction  of  the  tangent  and  normal  to  the  path  at  the 
time  t,  then  v  =  0m,  where  v  =  ds/dt  =  the  (scalar)  path-velocity,  and  dm  =  [(ds/R)]n, 
where  R  =  the  (scalar)  radius  of  curvature  of  the  path.  Then 

dv  dm       dv  »2 


Here  dv/dt  and  v2/R  are  the  familiar  expressions  for  the  components  of  acceleration  along 
the  tangent  and  normal. 


INDEX 


Abscissa,  173 
Absolute  value,  112 
Acceleration  9f  gravity,  73,  84 
Adding  machines,  97 
Addition,  algebraic,  112 

arithmetical,  88 

of  complex  quantities,  124 

of  vectors,  185 
Algebra,  elementary,  112-123 

of  complex  quantities,  124-127 

of  vectors,  18.5 
Alignment  charts,  179,  182 
Amortization  (sinking  fund),  67 
Analytical  geometry,  136-156 
Anchor  ring,  111 
Angles,  bisection  of,  102 

complementary,  128 

degrees  and  radians,  table,  44 

dihedral  and  solid,  110 

in  a  circle,  99 

in  analytical  geometry,  136 

in  trigonometry,  128-132 

minutes  and  seconds,  table,  69 

supplementary,  128 

units  of,  128 

Annuity  tables,  65,  67,  68 
Annulus,  area  of,  106 

contiguous  circles  in,  105 
Anti-friction  curve,  155 
Anti-gudermannian,  135 
Anti-hyperbolic  functions,  135 

graphs,  175;  series,  161 
Anti-togarithms,  92 
Anti-sines,  etc.,  132 

graphs,  174;  series,  161 
Apothecaries'  weight,  71 
Arc,  length  of  circular,  102,  106 
Archimedian  spiral,  154 
Arcsin,  etc.,  see  Anti-sines,  etc. 
Area,  units  of,  70,  76,  77 
Areas,  approximate  methods,  106,  170 

of  similar  figures,  99 

of  various  figures,  105 
Arithmetic,  88-98 
Arithmetical  mean,  115 

progression,  114 
Astroid,  153 
Asymptote,  of  hyperbola,  145,  146 

of  hyperbolic  spiral,  154 

of  tractrix,  155 


Ball-bearing  (annulus),  105 

Barrels,  volume  of,  110 

Baume  scale,  85 

Bessel's  formula,  121 

Binomial  coefficients,  tables  of,  39,  116 

series,  114 

theorem,  114 
Bisection,  of  a  line,  101 

of  an  angle,  102 
Bisectors  (in  triangle),  99,  134 
Board  measure,  71 


187 


Briggsian  logarithms,  113 
B.t.u.,  74,  75,  82 
Bushel,  70 


Calculating  machines,  98 
Calculus,  157-172 

rules  for  differentiation,  157 

table  of  integrals,  164 
Calendar,  83 
Cardioid,  153 
Casks,  volume  of,  110 
Catenary,  147 
Cavalieri's  theorem,  111 
Centesimal  measure  of  angles,  128 
Chaining  up  hill,  150 
Characteristic  of  logarithm,  92 

fractional,  94 
Charts,  alignment,  179,  182 

construction  of,  173 

contour  line,  179 

parallel  and  proportional,  184 
Circle,  constructions  for,  102-105 

equation  of,  137 

involute  of,  153 

tables  of  areas,  30,  32 
of  circumferences,  28,  32 
of  segments,  34,  35 

theorems  on  the,  99,  106 
Circles,  circumscribed,  99,  105,  134 

great,  on  a  sphere,  100 

inscribed,  99,  105,  134 

radical  axis  of,  100,  137 
Circular  measure  of  angles,  128 
table,  44 

mil,  70 
Cissoid,  155 

Coins,  value  of  foreign,  82 
Cologarithms,  93 
Combinations,  116 
Complex  quantities,  124-127 
Compound  interest  tables,  64,  66 
Computation,  graphical  methods  in,  174- 
185 

machines  for,  97 

numerical,  88 

Cones,  area  and  volume  of,  108,  109 
Conic  sections,  138-147 
Contour  line  charts,  179 
Conversion  tables,  74-82 
Coordinates,  polar,  137,  178 

rectangular,  136,  173 
Cosecant,  129;  tables,  51 

graph,  174 
Cosine,  129;  tables,  46,  52 

graph,  174;  series,  161 
Cotangent,  129;  tables,  47,  52 

graph,  174;  series,  161 
Coversed  sine,  129 
Cross-section  paper,  173,  178 
logarithmic,  176 
semi-logarithmic,  177 
Cube  In  geometry),  100,  110 


CUBE 


HYPERBOLIC  LOGARITHMS 


Cube  roots,  90 
of  1  ±  x,  91 

table  of,  16 

Cubes,  summation  of,  115 
table  of,  8 

Cubic  equation,  117 

Curvature,  163 

Curves,  empirical,  173 
of  various  functions,  174 
in  analytical  geometry,  151 

Cycloid,  151 

Cylinder,  area  and  volume,  107,  108 


Evolutes,  164 

Evolution,  in  algebra,  90 

Expansion  in  series,  114,  160 

Exponential  equations,  118 
function,  178 

graph,  174;  series,  160 

in  complex  algebra,  126,  127 

table,  57 

Exponents,  in  algebra,  113 
in  complex  algebra,  126,  127 

Exsecant,  129 

Extreme  and  mean  ratio,  102 


Decimal  equivalents,  tables,  33,  69 

point,  position  of,  89,  90 
Definite  integrals,  169 
Degrees,  and  minutes,  table,  69 

and  radians,  tables,  44,  45 
De  Moivre's  theorem,  126 
Denary  logarithms,  113 
Density,  81,  84 
Derivatives,  157,  158 

of  complex  quantities,  127 

of  definite  integrals,  170 

of  vectors,  186 

partial,  159 
Determinants,  123 
Diameters,  conjugate,  141,  146 
Differences,  115,  159 
Differential  calculus,  157-164 

equations,  171 
Dihedral  angles,  100 
Directrix,  of  catenary,  147 

of  ellipse,  140 

of  hyperbola,  144 

of  parabola,  138 
Division,  algebraic,  112 

arithmetical,  89 

by  logarithms,  93 

by  slide  rule,  95 

of  a  line,  101,  102 

of  complex  quantities,  124 
Distance  formula,  136 
Dodecahedron,  100,  110 
Dyne,  73 


Eccentric  angle,  141 
Eccentricity,  of  ellipse,  140 

of  hyperbola,  144 
Ellipse,  area  and  perimeter  of,  107 

constructions  for,  142 

properties  of,  140 
Ellipsoid,  volume  of,  110 
Elliptic  integrals,  170 
Empirical  curves,  173 
Energy,  units  of,  79,  80 
Epicycloid,  152 
Epitrochoid,  153 
Equations,  algebraic,  116 

differential,  171 

empirical,  174 

exponential,  118 

normal,  122 

simultaneous,  119,  121,  179 

solution  of,  by  trial,  118 

trigonometric,  118 

types  of,  116 
Errors,  absolute  and  relative,  88 

mean  square,  122 

probable,  121 ;  table,  63 


Factorials,  112 
Factoring,  in  algebra,  112 
Feet  and  inches,  table,  33 
Figures,  significant,  88 

similar,  99 

Financial  arithmetic,  98 
Focus,  of  ellipse,  140 

of  hyperbola,  144 

of  parabola,  138 
Force,  unit  of,  72,  74 
Fourier's  series,  162 
Fractions,  in  algebra,  112 
Frustum  of  cone,  108,  109 
Functions,  defined  by  integrals,  170 

graphs  of,  174 

implicit,  159 

of  a  complex  variable,  127 

of  two  variables,  159,  160,  178 


Gallon,  70 

Gamma  function,  170 
Geometrical  mean,  113,  115 
construction  for,  102 

progression,  115 
Geometry,  analytical,  136-156 

elementary,  99-111 
Golden  section,  102 
Gram,  71 

-calorie,  74,  75,  82 
Graphs,  empirical,  174 

in  computation,  174-185 

of  functions,  173 
Gravity,  acceleration  of,  73,  84 

specific,  84 
Gudermannian,  135 


Hardness,  scale  of,  85 
Harmonic  mean,  115 
Heat  units,  74,  75,  79,  80 
Helix,  156 
Hexagon,  103 
Horse-power,  73 
Huyghen's  approximation,  106 
Hyperbola,  area  of,  107 

as  type  of  power  function,  174 
conjugate,  146 
constructions  for,  147 
equilateral,  146 
properties  of,  144 
Hyperbolic  logarithms,  114 

table,  58 
sines,  etc.,  135 
graphs,  175 
series,  161 
tables,  60-62 
of  a  complex  variable,  127 


188 


HYPERBOLIC  LOGARITHMS 


POWER  FUNCTION 


Hyperbolic  (continued) 

spiral,  154 
Hypocycloid,  152 
Hypotrochoid,  153 


i  -  V^l7  125 
Icosahedron,  100,  110 
Identity,  116 
Imaginary  quantities,  124-127 

roots  of  equations,  118 
Implicit  functions,  159 
Inches,  and  feet,  table,  33 

and  millimeters,  table,  75 

miner's,  71 
Increment,  157 
Indeterminate  forms,  163 
Inflection-point,  160 
Integral  calculus,  164-170 
Integrals,  approximate  methods  for, 

definite,  169 

double,  170 

elliptic,  170 

probability,  170 

table  of,  164 
Integraph,  170 
Interest,  tables,  G4,  66 
Interpolation,  115 

in  logarithm  tables,  91,  92 
Intersection  of  lines,  102 
Inverse  sine,  etc.,  132 

graphs,  174;  series,  161 

sinh,  etc.,  135 

graphs,  175;  series,  161 
Involute,  164 

of  a  circle,  153 


Joule,  73 


Mass,  units  of,  77,  78 
Mathematics  (contents),  87 

tables  (contents),  1 
Maxima  and  minima,  159,  160 
Mean,  arithmetical,  115 

geometrical,  102,  113,  115 

harmonic,  115 

proportional,  113,  115 
construction  for,  102 

value  theorem,  169 
Measures,  weights  and,  70-85 
Medians  9f  a  triangle,  99,  134 
Mensuration,  105 
Metric  system,  72 
Minima  and  maxima,  160 
Minutes  and  seconds,  table,  69 
Mohs's  scale,  85 
Money,  foreign,  82 
Multiples,  of  IT,  table,  28 
170  of  0.4343  and  2.3026,  table,  62 

Multiplication,  algebraic,  112 

arithmetical,  89 

by  logarithms,  93 

by  slide  rule,  95 

of  complex  quantities,  124 

of  vectors,  186 

tables  (list  of),  89 


Napierian  logarithms,  114 

table,  58 
Natural  functions,  tables,  46,  52 

logarithms,  113;  table,  58 
Nomography,  179 
Normal  equations,  122 
Notation,  algebraic,  112 

by  powers  of  ten,  90 
Numerical  computation,  88 


Kilogram,  73 
Kilowatt,  73 


Latus  rectum,  of  ellipse,  140 
of  hyperbola,  145 
of  parabola,  138 
Least  squares,  121 
Lemniscate,  155 
Length,  units  of,  70,  74,  75 
Line,  equation  of,  136 

geometrical,  101 
Linear,  differential  equation,  171 

equation,  117 

function,  graph  of,  174 
Liter,  71 
Logarithmic  cross-section  paper,  176 

function,  173 

graph,  174;  series,  160 

spiral,  155 
Logarithms,  tables  (base  e),  58 

tables  (common),  40 

theory  of,  113 

use  in  computation,  91 
Lune,  110 


Maclaurin's  theorem,  161 
Mannheim  slide  rule,  96 
Mantissa  of  a  logarithm,  92 
negative,  93 


Obelisk,  volume  of,  109 
Octagon,  103 
Octahedron,  100,  110 
Ordinate,  173 
Orthocenter  of  a  triangle,  99 


Pappus,  theorems  of,  111 
Parabola,  area  of,  107 

constructions  for,  139 

properties  of,  138 

as  type  of  power  function,  174 
Paraboloid,  volume  of,  110 
Parallel  charts,  184 

lines,  101,  136 
Parallelogram,  area  of,  105 
Parameter,  137,  170 
Partial  derivatives,  159 
Permutations,  116 
Perpendicular  lines,  101,  136 
Peters's  formula,  121 
Planimeter,  170 
Polar  coordinates,  137,  178 

triangles  on  a  sphere,  101 
Polygons,  103;  table,  39 
Polyhedra,  100,  110 
Polynomial,  118 
Pound  and  poundal,  70,  73 
Power  function,  177 
graph,  174 

units  of,  73,  80,  81 


189 


POWERS 


TRUNCATED  PRISM 


Powers,  algebraic,  113 

arithmetical,  90 

by  logarithms,  93,  94 

by  slide  rule,  94 

in  complex  algebra,  125,  126,  127 

of  ten,  notation  by,  90 
Pressure,  units  of,  79,  80 
Prism,  area  and  volume,  107,  108 
Prismoidal  formula,  111 
Probability  integral,  170 
Probable  error,  121,  122 

table,  63 
Progression,  arithmetical,  114 

geometrical,  115 
Proportion,  in  algebra,  113 
Proportional  charts,  184 
Pyramids,  area  and  volume,  108,  109 


Quadrant,  128,  130 
Quadratic  equations,  117 
Quadrature,  170 
Quadrilateral,  area  of,  106 


Radian  measure  of  angles,  128 

tables,  44,  45 
Radical  axis,  100,  137 
Radicals  and  exponents,  113 
Ratio,  in  algebra,  113 

extreme  and  mean,  102 
Real  and  imaginary,  124 
Reciprocals,  90 

in  complex  algebra,  125 

of  1  ±  x,  90 

table  of,  24 

Rectangle,  area  of,  105 
Residuals,  121 

table,  63 

Rhombus,  area  of,  105 
Ribbon,  area  of,  106 
Roots,  see  Powers. 

of  an  equation,  116 


Sag,  in  the  catenary,  150 
Scalars,  185 
Schiele's  curve,  155 
Secant  of  an  angle,  129 

tables,  50,  52;  graph,  174 
Sector,  circular,  106 

spherical,  109 
Segments,  circular,  106 
tables,  34,  35 

of  paraboloid,  110 

spherical,  109;  table,  38 
Semi-logarithmic  paper,  177 
Series,  expansion  in,  114,  160 

Fourier  s,  162 

Maclaurin's  and  Taylor's,  161 

summation  of,  115 
Sexagesimal  measure,  128 
Significant  figures,  88 
Similar  figures,  99 
Simpson's  rule,  106,  111 
Simultaneous  equations,  119 
by  determinants,  123 
by  least  squares,  121 
contour  line  charts  for,  179 
Sine,  129;  tables,  46,  52 

graph,  174;  series,  161 
Sinking  fund  table,  67 


Slide  rule,  use  of,  94 

types  of,  97 
Slope,  157 
Solid  angle,  110 
Solids,  areas  and  volumes,  107 
Specific  gravity,  84 
Sphere,  area  and  volume,  109 

table  of  segments,  38 
of  volumes,  36 

theorems  on  the,  100,  109 
Spherical  segments,  109 
table,  38 

sector,  109;  wedge,  110 

triangles,  134 
area  of,  110,  134 
excess  of,  134 
Spheroid,  volume,  110 
Spiral,  hyperbolic,  154 

involute,  153 

logarithmic,  155 

of  Archimedes,  154 
Square  roots,  90 
of  1  ±  x,  90 
table  of,  12 
Squares,  summation  of,  115 

table  of,  2 

Steradian,  steregon,  110 
Submultiples,  124 
Subnormal,  subtangent,  138 
Subtraction,  algebraic,  112 

arithmetical,  88 

of  complex  quantities,  124 

of  logarithms,  93 

of  vectors,  185 
Summation  of  series,  115 
Surface  for  /  (x,  y,}  178 
Surfaces,  areas  of,  107 
Symbols,  algebraic,  112 
Symmetrical  triangles,  101 


Tables,  list  of,  1 
Tangent  of  an  angle,  129 

graph,  174;  series,  161 

tables,  48,  52 
to  circle,  99 

construction  of,  103 
Tape,  sag  of,  150 
Taylor's  theorem,  161 
Tens,  notation  by,  90 
Tetrahedron,  100,  110 
Therm,  74 

Three-halves  powers,  20,  22 
Time,  83 

Torus,  area  and  volume,  111 
Tractrix,  155 
Trapezoid,  area  of,  105 
Trial  and  error  method,  118 
Triangles,  plane,  99,  105,  134 

solution  of,  132 
polar,  101 
spherical,  110,  134 

solution  of,  134 
Trigonometric  equations,  118 
functions,  128 

graphs,  174;  series,  161 

of  a  complex  variable,  127 

tables,  44,  46,  52 
Trigonometry,  128-135 
Trochoid,  152 
Troy  weight,  71 
Truncated  prism,  107,  108 


190 


UNGULA  ZONE 

Ungula,  108  Watt,  73 

Units,  72  Wedge,  109 

spherical,  119 

Weight,  units  of,  71,  77,  78 

Variable,  change  of,  169  Weights  and  measures,  70-85 

Vector  analysis,  185  Work,  units  of,  73,  79,  80 

Velocity,  units  of,  78,  80 

Versed  sine,  129;  graph,  174  Yard,  70 

Volume,  units  of,  70,  71,  76,  77  Year,  83 

Volumes,  of  solids,  107 

of  similar  figures,  99  Zone  of  a  sphere,  109 


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